The Origin of Homochirality by Rotational Magnetoelectrochemistry

Abstract

The origin of homochirality by rotational magnetoelectrochemistry was theoretically examined. Electrochemical reductions in a rotating solution under a static vertical magnetic field were concluded to yield microscopic vortices with L-activity for enantiomeric reagents, whereas D-active vortices arise from electrochemical oxidation. The reduction case was experimentally verified by rotational magnetoelectrodeposition (RMED) of copper films using an electrolysis cell rotating in a magnetic field, where L-active screw dislocations were created by L-active microscopic vortices. In all the cases of the directions of magnetic polarity and system rotation, the RMED films exhibited L-activity for the enantiomeric reactions of amino acids.

1. Introduction

Chirality is one of the fundamental issues in biochemical systems, where a number of chemical reactions are controlled by chiral stereoselectivity, and symmetry-breaking in chirality concerning the birth of life has long intrigued many scientists. Chiral symmetry breaking occurs when a physical or chemical process that does not have a preference for the production of one or the other enantiomer spontaneously generates a large excess of one of the two enantiomers: lefthanded (L) or righthanded (D). From the energetic point of view, these two enantiomers can exist with an equal probability, and inorganic processes involving chiral products commonly yield a racemic mixture of both (L) and (D) enantiomers. However, life on earth utilizes only one type of amino acids and only one type of natural sugars: (L)-amino acids and (D)-sugars [1].

One of the most interesting questions in the molecular evolution towards the birth of life is how the biological system selected one enantiomer of a chiral biomolecule. To identify possible external influences on such a selection, various mechanisms have been proposed, such as circularly polarized light [2,3,4], magneto-chiral fields [5], vortex and stirring conditions [6,7,8], nonlinear autocatalysis [9,10], gravitational effect [11], and combinations of external fields [12]. However, which one is chosen, D- or L-activity, still depends on the accidental nature.

Fluid flow, such as vortex motion, could also be one of the most probable symmetry-breaking factors. Kondepudi et al. reported that solution stirring breaks the racemic state in the crystallization of enantiomorphous NaClO₃ [6]. Ribo et al. reported that mechanical vortices induce chirality in the aggregation of achiral porphyrins in aqueous solutions [7]. These facts demonstrate that macroscopic vortices have an asymmetric influence on crystallization and aggregation at the nanometer scale.

Furthermore, under the prebiotic oceans on the early Earth, surfaces of minerals could serve as catalysts in the formation of amino acids [13,14]. If such catalytic surfaces were chiral, the chiral symmetry would be broken for the enantiomers of amino acids, and one-handed amino acids would become dominant. It is a crucial point for the formation of a homochiral biosystem whether the minerals have chiral surfaces or not. Thus, studies of chiral surface formation in deposition and crystal growth processes are important to explore the origin of life.

According to F. C. Frank et al. [15,16], a crystal surface has numerous screw dislocations, of which steps spirally proceed with the addition of atoms or molecules as the sites of growth. As exhibited by STM observation, in electrodeposition also, a lot of screw dislocations emerge on the growing surfaces [17], and the center of a screw dislocation has a typical chiral structure, which could serve as a chiral catalyst. As will be shown below, in electrodeposition under a vertical magnetic field, Lorentz force induces microscopic magnetohydrodynamic (MHD) vortices, which control the formation of chiral screw dislocations, adding chiral activities to the deposit surface. As discussed in Appendix E, the MHD vortices may bias or enhance existing tendencies rather than solely cause such structures.

In an electrode reaction under a vertical magnetic field, a tornado-like macroscopic rotation called a vertical magnetohydrodynamic (MHD) flow (VMHDF) appears over the electrode [18,19,20,21]. Under the rotation, numerous microscopic vortices emerge so that as shown in Figure 1, the electrodeposition under a vertical magnetic field (magnetoelectrodeposition, MED) proceeds with vortices of characteristic sizes in accordance with the nucleation in three generations; in the 1st generation, 2D nucleation with micro-MHD vortices of a representative size of 10⁻⁴ m, and in the 2nd generation, 3D nucleation with nano-MHD vortices of a size of 10⁻⁷ m, and finally in the 3rd generation, screw dislocation with ultra-micro-MHD vortices of 10⁻¹⁰ m. All the generations form a nesting box structure so that micro-MHD vortices result from VMHDF, nano-MHD vortices from micro-MHD vortices, and ultra-micro-MHD vortices from nano-MHD vortices [18,19]. The direction of VMHDF depends on the magnetic field polarity. Hence, the chiral sign of the deposit surface becomes opposite when the magnetic field is reversed, representing odd chirality for magnetic field polarity [20]. The precessional rotation from higher generation is transmitted to lower generation through the instability processes of nucleation so that the resultant chiral activities are sensitive to the electrochemical conditions of each nucleation process. Consequently, the enantiomeric activity of the deposit surface varies not only with the direction of the magnetic field but also in the presence of electrochemical additives, such as chloride ions [21].

In an electrochemical reaction under a magnetic field, an inviscid layer of ionic vacancies is formed on the electrode surface [22,23]. Ionic vacancies are produced from the conservation of linear momentum and electric charge in the charge transfer during an electrode reaction [24,25]. Unstable embryo vacancies immediately after their creation are rapidly stabilized by solvation and develop into solvated vacancies surrounded by ionic clouds. Figure 2 exhibits the schematics of solvated negative and positive ionic vacancies yielded at the cathode and the anode, respectively. The solvation energy of the ionic cloud is used for the expanding work of the vacancy core, saving the energy in it, so that an ionic vacancy behaves as an iso-entropic particle without entropy production. Therefore, ionic vacancies do not interact with other solvent molecules, keeping a natural lifetime of 1 s [26,27], which is extraordinarily longer than a thermal collision period of 10⁻¹⁰ s. This means that an ionic vacancy acts as an atomic-scale lubricant and that the vacancy layer behaves like an inviscid fluid. Due to quite a high Reynolds number, the microscopic vortices could not rotate in a usual solution but rotated in or on a vacancy layer.

Under a vertical magnetic field, microscopic anticlockwise (ACW) and clockwise (CW) MHD vortices arise on and in a vacancy layer, which cannot be discriminated as they are. However, as shown in Figure 3a, to conserve local angular momentum and mass, we can assume a pair of vortices rotates, in opposite directions with upward and downward flows. As a result, the combination of the vortex pair is limited to two cases, i.e., ACW and CW vortices with upward and downward flows or downward and upward flows, respectively [18,19]. As shown in Figure 3b,c, an upward vortex sucks up and gathers ionic vacancies, whereas a downward one drives them away, so that the former rotates on the free surface covered with ionic vacancies and the latter rotates on the exposed rigid surface. Furthermore, as shown in Figure 4, because the active reaction sites among ionic vacancies rotate with a free-surface vortex (FV), any chirality would not take place. Chiral reaction product such as chiral screw dislocation is thus expected only by a rigid-surface vortex (RV) rotating against stationary active sites.

Figure 5 shows a screw-dislocation-like deposition theoretically calculated for 3D nucleation formed by a clockwise (CW) nano-MHD vortex on the rigid surface in the 2nd generation [19]. Though much larger scale of length than actual screw dislocation, the same CW chirality as the micro-MHD vortex appears. This result indicates that chiral screw dislocations are created by ultra-micro MHD vortex flows on the rigid surface with the same chirality, which would be generalized as a fact that a chiral nucleus is created by a corresponding chiral vortex on the rigid surface. If unrelated to electrode reactions, chiral vortices were made, and we could control the chiral activity of a deposit surface by the external physical parameters, including a magnetic field.

As discussed above, for controlling chiral microscopic vortices unrelated to chemical or electrochemical conditions, it is insufficient to use only a magnetic field as an external physical parameter. We therefore introduce a new physical parameter, the angular velocity of system rotation. As will be shown later, by using such two parameters, we succeeded in producing the microscopic MHD vortices with controlled chirality.

The theoretical outline of the present paper is therefore provided as follows; we first assume an electrolysis system without a VMHDF under a vertical magnetic field, where only microscopic vortices called MHD vortex flows are induced by the Lorentz force fluctuation, and to conserve local angular momentum and mass, a pair of neighboring vortices appear, rotating each other in the opposite directions. We then apply a horizontal rotation to the system, thereby imposing the Coriolis force on the vortices to induce precession. As a result, from the balance between the Lorentz force and the Coriolis force, either of the vortex pairs tends to stop rotation so that the other can select the precession from the system rotation. As a result, independent of the directions of the magnetic field and rotation, only a pair of ACW and CW vortices on the rigid and free surfaces is permitted, respectively. Since chirality arises only from the rigid-surface vortices, in the case of rotational magnetoelectrodeposition (RMED), L-active, ACW screw dislocation would be obtained. If an enantiomeric reagent had the sensitivity for vortex rotation, an L-active reaction would be predominant. The experimental validation is presented by the chiral activity of copper RMED. Therefore, the theoretical procedure will be taken in accordance with the case of RMED.

2. Theory

2.1. Basic Equations

First, we explicitly consider an inertial frame with a static magnetic field. In this frame, a coordinate system (x, y, z) defined for an electrolysis cell containing a planar electrode on the x-y plane, rotates with the cell at an angular velocity $\overrightarrow{\mathsf{\Omega }}$ (s⁻¹) around the z-axis (see Figure 6). As will be shown later, in the present case, we assume that the electrolysis current flows perfectly parallel to the magnetic field so that a tornado-like macroscopic rotation (VMHDF) arising from the macroscopic distortion of current lines does not occur [22].

As shown in Appendix A, the overall current density $\overrightarrow{J}$ (A m⁻²) flows under a magnetic flux density $\overrightarrow{B}$ (T, tesla), so that the Lorentz force per unit mass, i.e., acceleration (m s⁻²) is generated in the following,

$${\overrightarrow{F}}_{\mathrm{L}}=\overrightarrow{J}\times \overrightarrow{B}$$

(A4)

In addition, an observer at rest in the rotating system recognizes two kinds of forces, i.e.,

$${\overrightarrow{F}}_{\mathrm{R}}=2\overrightarrow{\mathsf{\Omega }}\times \overrightarrow{u}-{\displaystyle \frac{1}{2}}\nabla \left({\left|\overrightarrow{\mathsf{\Omega }}\times \overrightarrow{r}\right|}^2\right)$$

(A7)

where ${\overrightarrow{F}}_{\mathrm{R}}$ is rotational force per unit mass, i.e., acceleration (m s⁻²), $\overrightarrow{\mathsf{\Omega }}$ is the vector of angular velocity (s⁻¹), $\overrightarrow{u}$ is the vector of velocity (m s⁻¹), and $\overrightarrow{r}$ is the vector of position (m). The first and second terms on the right-hand side represent the Coriolis force and the centrifugal force.

Then, we consider an incompressible fluid at a uniform temperature so that the basic equations are given in the following: the momentum equation is first in tensor notation,

$${\displaystyle \frac{\partial u_{\mathrm{i}}}{\partial t}}+u_{\mathrm{j}}{\displaystyle \frac{\partial u_{\mathrm{i}}}{\partial x_{\mathrm{j}}}}-{\displaystyle \frac{B_{\mathrm{j}}}{\rho {\mu }_0}}{\displaystyle \frac{\partial B_{\mathrm{i}}}{\partial x_{\mathrm{j}}}}=2{\epsilon }_{\mathrm{i}\mathrm{j}\mathrm{k}}{u}_{\mathrm{j}}{\mathsf{\Omega }}_{\mathrm{k}}+\mathsf{\nu }{\nabla }^2{u}_{\mathrm{i}}-{\displaystyle \frac{\partial }{\partial x_{\mathrm{i}}}}\left({\displaystyle \frac{P}{\rho }}+{\displaystyle \frac{{\left|\overrightarrow{B}\right|}^2}{2{\mu }_0\rho }}-{\displaystyle \frac{1}{2}}{\left|\overrightarrow{\mathsf{\Omega }}\times \overrightarrow{r}\right|}^2\right)$$

(A8)

where $u_{\mathrm{i}}$ is the velocity component (m s⁻¹) (i = 1, 2, 3), and the coordinate $\left(x,y,z\right)$ (m) is expressed by $\left(x_1,x_2,x_3\right).$ ${\nabla }^2$ implies ${\partial }^2/\partial x_1^2+{\partial }^2/\partial x_2^2+{\partial }^2/\partial x_3^2$, and ${\mu }_0$ is the magnetic permeability (4$\pi \times {10}^{-7}$ N A⁻²). $\mathsf{\nu }$ and $\rho$ are the kinematic viscosity (m² s⁻¹) and the density (kg m⁻³), respectively. ${\epsilon }_{\mathrm{i}\mathrm{j}\mathrm{k}}$ is Levi Civita symbol (transposition of tensor). In view of an incompressible fluid, the continuity equation is fulfilled.

$${\displaystyle \frac{\partial u_{\mathrm{i}}}{\partial x_{\mathrm{i}}}}=0$$

(A9)

If a fluid element has a velocity $\overrightarrow{u}$, the electric field it will experience is not $\overrightarrow{E}$ (V) as measured by a stationary observer, but $\overrightarrow{E}+\overrightarrow{u}\times \overrightarrow{B}$. In an electrolytic solution, the electricity is carried by the diffusion as well as conductivity of ionic species so that the current density will be given by

$$\overrightarrow{J}={\sigma }^{\mathrm{*}}\left(\overrightarrow{E}+\overrightarrow{u}\times \overrightarrow{B}\right)-F\sum _{\mathrm{i}}{z}_{\mathrm{i}}{D}_{\mathrm{i}}\nabla C_{\mathrm{i}}$$

(A10)

where $\nabla$ is defined by $\left(\partial /\partial x_1,\partial /\partial x_2,\partial /\partial x_3\right).$ ${\sigma }^{\mathrm{*}}$ is the electrical conductivity (S m⁻¹) defined by

$${\sigma }^{\mathrm{*}}=F^2\sum _{\mathrm{i}}{z}_{\mathrm{i}}^2{\lambda }_{\mathrm{i}}^{\mathrm{*}}{C}_{\mathrm{i}}$$

(A11)

where $z_{\mathrm{i}}$ is the signed charge number, ${\lambda }_{\mathrm{i}}^{*}$ is the mobility (m² V⁻¹ s⁻¹), F is the Faraday constant (96,500 C mol⁻¹), $C_{\mathrm{i}}$ is the molar concentration of ionic species i (mol m⁻³), and $D_{\mathrm{i}}$ is the diffusion constant (m² s⁻¹).

Applying Equation (A10) to Maxwell’s equations, Equations (A1)–(A3), in Appendix A, we finally obtain the equation of magnetic flux density in tensor notation.

$${\displaystyle \frac{\partial B_{\mathrm{i}}}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_{\mathrm{j}}}}\left(u_{\mathrm{j}}{B}_{\mathrm{i}}-u_{\mathrm{i}}{B}_{\mathrm{j}}\right)={\eta \nabla }^2\overrightarrow{B}$$

(A17)

where $\eta$ is the resistivity (magnetic viscosity) (m² s⁻¹) defined by

$$\eta \equiv {\displaystyle \frac{1}{{\sigma }^{\mathrm{*}}{\mu }_0}}$$

(A15)

Finally, the mass transfer equation of metallic ion is expressed by the concentration $C_{\mathrm{m}}$ and diffusion coefficient $D_{\mathrm{m}}$.

$${\displaystyle \frac{\partial C_{\mathrm{m}}}{\partial t}}+\left(\overrightarrow{u}·\nabla \right)C_{\mathrm{m}}=D_{\mathrm{m}}{\nabla }^2{C}_{\mathrm{m}}$$

(A18)

2.2. Equations of the Nonequilibrium Fluctuations

As discussed in Appendix B, we simply consider an infinite horizontal layer of fluid in a state of uniform rotation with an angular velocity and subject to a uniform magnetic flux density. Here, we shall restrict our discussion of this problem to the case when $\overrightarrow{B}$ and $\overrightarrow{\mathsf{\Omega }}$ act in the same vertical direction. In addition, the current density $\overrightarrow{J}$ also works in the vertical direction. Considering that a vector of rotation is an axial vector, we write down the following notation.

$$\overrightarrow{\mathsf{\Omega }}=\left(\mathrm{0,0},\mathsf{\Omega }\right)$$

(B16a)

$$\overrightarrow{B}=\left(\mathrm{0,0},B_0\right)$$

(B16b)

$$\overrightarrow{J}=\left(\mathrm{0,0},J_z\right)$$

(B16c)

As a result, for each generation of nucleation, we obtain the fluctuation equations.

$${\displaystyle \frac{\partial b_{\mathrm{z}}}{\partial t}}={\eta \nabla }^2{b}_{\mathrm{z}}+B_0{\displaystyle \frac{\partial w}{\partial z}}$$

(B17a)

$${\displaystyle \frac{\partial j_{\mathrm{z}}}{\partial t}}=\eta {\nabla }^2{j}_{\mathrm{z}}+{\displaystyle \frac{B_0}{{\mu }_0}}{\displaystyle \frac{\partial {\omega }_z}{\partial z}}$$

(B17b)

$${\displaystyle \frac{\partial {\omega }_{\mathrm{z}}}{\partial t}}=\mathsf{\nu }{\nabla }^2{\omega }_{\mathrm{z}}+{\displaystyle \frac{B_0}{\rho }}{\displaystyle \frac{\partial j_{\mathrm{z}}}{\partial z}}+2\mathsf{\Omega }{\displaystyle \frac{\partial w}{\partial z}}$$

(B17c)

$${\displaystyle \frac{\partial }{\partial t}}{\nabla }^{2}w=\mathsf{\nu }{\nabla }^{4}w+{\displaystyle \frac{B_0}{\rho {\mu }_0}}{\displaystyle \frac{\partial }{\partial z}}{\nabla }^2{b}_{\mathrm{z}}-2\mathsf{\Omega }{\displaystyle \frac{\partial {\omega }_{\mathrm{z}}}{\partial z}}$$

(B17d)

where $b_{\mathrm{z}}$ and $j_{\mathrm{z}}$ are the z-fluctuation components of the magnetic flux density (T) and the current density (A m⁻²). $w$ and ${\omega }_{\mathrm{z}}$ are the z-components of the fluctuations of the velocity (m s⁻¹) and the vorticity (s⁻¹).

The concentration of the metallic ion is expressed as

$$C_{\mathrm{m}}=C_{\mathrm{m}}^{*}+c_{\mathrm{m}}$$

(B18)

where $C_{\mathrm{m}}^{*}$ and $c_{\mathrm{m}}$ are the molar concentration in the absence of fluctuation (mol m⁻³) and the concentration fluctuation (mol m⁻³), respectively. Mass transfer equation, Equation (A18), is also rewritten as

$${\displaystyle \frac{\partial c_{\mathrm{m}}}{\partial t}}+w{L}_{\mathrm{m}}={D_{\mathrm{m}}\nabla }^2{c}_{\mathrm{m}}$$

(B19)

$L_{\mathrm{m}}$ is the average concentration gradient of the metallic ion in each generation of nucleation (mol m⁻⁴). It takes different values depending on the generations; the concentration gradient of the 1st generation is formed in the diffuse layer of the electric double layer, that of the 2nd generation is defined by the diffusion layer, and in the 3rd generation, it is formed in front of screw dislocations. However, because they are connected in series, assuming that their formations are controlled by the mass transfer in the diffusion layer of the 2nd generation, we can simply describe their expressions by the ratio of the concentration difference ${\theta }_{\infty }^{*}$ (mol m⁻³) against the average diffusion layer thickness $〈{\delta }_{\mathrm{c}}〉$ (m) in the following:

$$L_{\mathrm{m}} \equiv {\displaystyle \frac{{\theta }_{\mathrm{\infty }}^{*}}{〈{\delta }_{\mathrm{c}}〉}}$$

(B20a)

${\theta }_{\infty }^{*}$ is defined by

$${\theta }_{\mathrm{\infty }}^{*}\equiv C_{\mathrm{m}}^{*}\left(z=\mathrm{\infty }\right)-C_{\mathrm{m}}^{*}\left(z=0\right)$$

(B20b)

where $C_{\mathrm{m}}^{*}\left(z=\infty \right)$ and $C_{\mathrm{m}}^{*}\left(z=0\right)$ are the bulk and surface concentration (mol m⁻³) of the metallic ion, respectively.

2.3. Amplitude Equations of the Fluctuations

For the fluctuations, we assume the following 2D plane waves.

$$w=W^0\left(z,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\left(k_{\mathrm{x}}x+k_{\mathrm{y}}y\right)\right]$$

(C1a)

$${\omega }_{\mathrm{z}}={\mathsf{\Omega }}^0\left(z,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\left(k_{\mathrm{x}}x+k_{\mathrm{y}}y\right)\right]$$

(C1b)

$$b_{\mathrm{z}}={\mathrm{K}}^0\left(z,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\left(k_{\mathrm{x}}x+k_{\mathrm{y}}y\right)\right]$$

(C1c)

$$j_{\mathrm{z}}=J^0\left(z,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\left(k_{\mathrm{x}}x+k_{\mathrm{y}}y\right)\right]$$

(C1d)

$$c_{\mathrm{m}}={\mathsf{\Theta }}^0\left(z,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\left(k_{\mathrm{x}}x+k_{\mathrm{y}}y\right)\right]$$

(C1e)

where ‘i’ is the unit imaginary number. ${\mathrm{W}}^0\left(z,t\right)$, ${\mathsf{\Omega }}^0\left(z,t\right)$, ${\mathrm{K}}^0\left(z,t\right)$, ${\mathrm{J}}^0\left(z,t\right),$ and ${\mathsf{\Theta }}^0\left(z,t\right)$ are the amplitudes of the fluctuations, and $k_{\mathrm{x}}$ and $k_{\mathrm{y}}$ are the wavenumbers (m⁻¹) in the x- and y-directions, respectively.

Substituting Equations (C1a)–(C1e) into Equations (B17a)–(B17d) and (B19), and assuming that the fluctuations are at a quasi-steady state, we obtain

$$\left({\mathrm{D}}^2-k^2\right){\mathrm{K}}^0=-\left({\displaystyle \frac{B_0}{\eta }}\right)\mathrm{D}{W}^0$$

(C3a)

$$\left({\mathrm{D}}^2-k^2\right)J^0=-\left({\displaystyle \frac{B_0}{{\mu }_0\eta }}\right)\mathrm{D}{\mathsf{\Omega }}^0$$

(C3b)

$$\left({\mathrm{D}}^2-k^2\right){\mathsf{\Omega }}^0=-\left({\displaystyle \frac{B_0}{\rho \mathsf{\nu }}}\right)\mathrm{D}{J}^0-\left({\displaystyle \frac{2\mathsf{\Omega }}{\mathsf{\nu }}}\right)\mathrm{D}{W}^0$$

(C3c)

$${\left({\mathrm{D}}^2-k^2\right)}^2{W}^0=-\left({\displaystyle \frac{B_0}{{\mu }_0\rho \mathsf{\nu }}}\right)\mathrm{D}\left({\mathrm{D}}^2-k^2\right){\mathrm{K}}^0+\left({\displaystyle \frac{2\mathsf{\Omega }}{\mathsf{\nu }}}\right)\mathrm{D}{\mathsf{\Omega }}^0$$

(C3d)

$$\left({\mathrm{D}}^2-k^2\right){\mathsf{\Theta }}^0=\left({\displaystyle \frac{L_{\mathrm{m}}}{D_{\mathrm{m}}}}\right)W^0$$

(C3e)

where $\mathrm{D}\equiv \mathrm{d}/\mathrm{d}z$ and $k\equiv {\left(k_{\mathrm{x}}^2+k_{\mathrm{y}}^2\right)}^{1/2}$. Substituting Equation (C3b) into Equation (C3c), and using the resistivity in Equation (A15), we obtain

$$\left\{{\left({\mathrm{D}}^2-k^2\right)}^2-Q^{*}{\mathrm{D}}^2\right\}{\mathsf{\Omega }}^0=-T^{*}\mathrm{D}\left({\mathrm{D}}^2-k^2\right)W^0$$

(C4a)

where $Q^{*}$ and $T^{*}$ imply the magneto-induction coefficient (m⁻²) and Coriolis force coefficient (m⁻²), expressed as

$$Q^{*}\equiv {\displaystyle \frac{{\sigma }^{*}{B}_0^2}{\rho \nu }}$$

(C4b)

and

$$T^{*}\equiv {\displaystyle \frac{2\mathsf{\Omega }}{\nu }}$$

(C4c)

Then, substitution of Equation (C3a) into Equation (C3d) leads to

$$\left\{{\left({\mathrm{D}}^2-k^2\right)}^2-Q^{*}{\mathrm{D}}^2\right\}{W}^0=T^{*}\mathrm{D}{\mathsf{\Omega }}^0$$

(C5)

Introducing a representative length d (m), we have Tayler number.

$$T\equiv {T^{*}}^2{d}^4$$

(C6a)

In addition, the nondimensional form of $Q^{*}$ is given by

$$Q\equiv {\displaystyle \frac{{\sigma }^{*}{B}_0^2{d}^2}{\rho \nu }}\left(=Q^{*}{d}^2\right)$$

(C6b)

Both hand sides of Equation (C4a) are multiplied with $\left\{{\left({\mathrm{D}}^2-a^2\right)}^2-Q^{*}{\mathrm{D}}^2\right\}$, and Equation (C5) is substituted into the resultant equation, so that we derive the final equation to solve.

$$\left[{\left\{{\left({\mathrm{D}}^2-a^2\right)}^2-Q^{*}{\mathrm{D}}^2\right\}}^2+{T^{*}}^2{\mathrm{D}}^2\left({\mathrm{D}}^2-k^2\right)\right]{\mathsf{\Omega }}^0=0$$

(C7)

Equations (C5) and (C7) with Equation (C3e) are solved under the following conditions.

2.4. Boundary Conditions of the Fluctuations on the Rigid and Free Surfaces

The fluid of the solution is confined between $z=0$ and $\mathrm{\infty }$. Regardless of the nature of these boundary surfaces, we must first require that the vertical velocity be zero.

$$w=0 \mathrm{for} z=0 \mathrm{and} \mathrm{\infty }$$

(1a)

where the condition $w=0$ for $z=0$ also means

$${\displaystyle \frac{\partial w}{\partial x}}={\displaystyle \frac{\partial w}{\partial y}}=0$$

(1b)

At the electrode surface $z=0$, as shown in Figure 3, two types of vortices rotate on the rigid and free surfaces. Accordingly, we distinguish two types of hydrodynamic boundary surfaces.

2.4.1. Rigid Surface on Which No Slip Occurs

The conditions that no slip occurs on this surface imply that not only w, but also the horizontal x- and y-components of the velocity u and v (m) vanish.

$$u=v=0 \mathrm{in} \mathrm{addition} \mathrm{to} w=0 \text{on a rigid surface}$$

(2)

These conditions must be satisfied for all x and y on the surface, so that for the rigid surface, we obtain

$${\displaystyle \frac{\partial u}{\partial x}}={\displaystyle \frac{\partial v}{\partial y}}=0$$

(3a)

$${\displaystyle \frac{\partial v}{\partial x}}={\displaystyle \frac{\partial u}{\partial y}}=0$$

(3b)

Substituting Equation (1b) into the equation of continuity in Equation (A9), we can derive the boundary condition of $w.$

$${\displaystyle \frac{\partial w}{\partial z}}=0 \text{on the rigid surface}$$

(4)

The normal component ${\omega }_{\mathrm{z}}$ of the vorticity is defined by

$${\omega }_{\mathrm{z}}={\displaystyle \frac{\partial v}{\partial x}}-{\displaystyle \frac{\partial u}{\partial y}}$$

(5)

From Equations (3b) and (5), we obtain the boundary condition of ${\omega }_z$.

$${\omega }_{\mathrm{z}}=0 \text{on the rigid surface}$$

(6)

2.4.2. Free Surface Where No Tangential Stress Acts

The conditions on a free surface are that the stress tensors are zero, i.e.,

$$P_{\mathrm{x}\mathrm{z}}=P_{\mathrm{y}\mathrm{z}}=0$$

(7)

where $P_{\mathrm{x}\mathrm{z}}$ and $P_{\mathrm{y}\mathrm{z}}$ are the viscous stress tensor (Pa):

$$P_{\mathrm{x}\mathrm{z}}={\mu }^{*}\left({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}}\right)$$

(8a)

and

$$P_{\mathrm{y}\mathrm{z}}={\mu }^{*}\left({\displaystyle \frac{\partial v}{\partial z}}+{\displaystyle \frac{\partial w}{\partial y}}\right)$$

(8b)

where ${\mu }^{*}$ is the viscosity (Pa s). Substitution of Equation (1b) into Equation (7) leads to

$${\displaystyle \frac{\partial u}{\partial z}}={\displaystyle \frac{\partial v}{\partial z}}=0$$

(8c)

Differentiating the equation of continuity in Equation (A9) regarding z, and substituting Equation (8c) into the resulting equation, we finally get the boundary condition of $w$.

$${\displaystyle \frac{{\partial }^{2}w}{\partial z^2}}=0 \text{on the free surface}$$

(9)

Then, after the differentiation of Equation (5) with respect to z, substituting for Equation (8c) in the resulting equation, we have

$${\displaystyle \frac{\partial {\omega }_z}{\partial z}}=0 \text{on the free surface}$$

(10)

We also add the following condition.

$${\omega }_z=0 \mathrm{for} z=\mathrm{\infty }$$

(11)

In addition to these pure hydrodynamic conditions, we can write down the other conditions concerning mass transfer. From Fick’s first law, we have the following relationship between the fluctuations of the current density and concentration

$$j_z=-z_{\mathrm{m}}F{D}_{\mathrm{m}}{\left({\displaystyle \frac{\partial c_{\mathrm{m}}}{\partial z}}\right)}_{z=0}$$

(12)

together with the conditions

$$c_{\mathrm{m}}=0 \mathrm{for} z=\mathrm{\infty }$$

(13)

Based on the above discussions, the boundary conditions of the amplitudes are summarized as follows.

$$W^0={\mathsf{\Omega }}^0={\mathsf{\Theta }}^0=0 \mathrm{for} z=\mathrm{\infty }$$

(14)

$$W^0=0 \mathrm{for} z=0$$

(15)

$$J^0=-z_{\mathrm{m}}F{D}_{\mathrm{m}}\mathrm{D}{\mathsf{\Theta }}^0 \mathrm{for} z=0$$

(16)

and either

$${\mathsf{\Omega }}^0=\mathrm{D}{W}^0=0 \mathrm{for} z=0 \text{on the rigid surface}$$

(17a)

or

$$\mathrm{D}{\mathsf{\Omega }}^0={\mathrm{D}}^2{W}^0=0 \mathrm{for} z=0 \text{on the free surface}$$

(17b)

2.5. General Solution of ${\Omega }^0$

First, to obtain the general solution of ${\mathsf{\Omega }}^0$ in Equation (C7), as shown in Appendix D, the following function form is assumed.

$${\mathsf{\Omega }}^0=\mathrm{f}\left(z,t\right){\mathrm{e}}^{-kz}\hspace{1em}\mathrm{for} 0\le kz\ll 1$$

(D1)

For the condition $0\le kz\ll 1$, ${\mathsf{\Omega }}^0$ mainly follows $\mathrm{f}\left(z,t\right)$, which is expressed by a series of z.

$$\mathrm{f}\left(z,t\right)=a_0+a_{1}z+a_2{z}^2+\cdots \cdots \cdots ={\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}{a}_{\mathrm{i}-1}{z}^{\mathrm{i}-1}$$

(D2)

where $a_{0,}$ $a_{1,}$ $a_2$, $\cdots \cdots \cdots$ are the arbitrary quasi-static constants, which are functions of t.

Substituting Equation (D1) into Equation (C7), we finally derive the following equation.

$$\begin{gathered} {\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}{\left[{\displaystyle \frac{\left(\mathrm{i}+7\right)!}{\left(\mathrm{i}-1\right)!}}\right.a}_{\mathrm{i}+7}-{\displaystyle \frac{\left(\mathrm{i}+6\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_7{a}_{\mathrm{i}+6}+{\displaystyle \frac{\left(\mathrm{i}+5\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_6{a}_{\mathrm{i}+5}-{\displaystyle \frac{\left(\mathrm{i}+4\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_5{a}_{\mathrm{i}+4}+{\displaystyle \frac{\left(\mathrm{i}+3\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_4{a}_{\mathrm{i}+3}-{\displaystyle \frac{\left(\mathrm{i}+2\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_3{a}_{\mathrm{i}+2} \\ +{\displaystyle \frac{\left(\mathrm{i}+1\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_2{a}_{\mathrm{i}+1}-{\displaystyle \frac{\mathrm{i}!}{\left(\mathrm{i}-1\right)!}}{\alpha }_1{a}_{\mathrm{i}}+\left.{\alpha }_0{a}_{\mathrm{i}-1}\right]z^{\mathrm{i}-1}=0 \end{gathered}$$

(D6)

Equation (D6) must be an identity with regard to z, so that each i-th term is equal to zero.

$$\begin{gathered} {\displaystyle \frac{\left(\mathrm{i}+7\right)!}{\left(\mathrm{i}-1\right)!}}{a}_{\mathrm{i}+1}-{\displaystyle \frac{\left(\mathrm{i}+6\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_7{a}_{\mathrm{i}+6}+{\displaystyle \frac{\left(\mathrm{i}+5\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_6{a}_{\mathrm{i}+5}-{\displaystyle \frac{\left(\mathrm{i}+4\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_5{a}_{\mathrm{i}+4}+{\displaystyle \frac{\left(\mathrm{i}+3\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_4{a}_{\mathrm{i}+3}-{\displaystyle \frac{\left(\mathrm{i}+2\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_3{a}_{\mathrm{i}+2} \\ +{\displaystyle \frac{\left(\mathrm{i}+1\right)!}{\left(\mathrm{i}-1\right)!}}{\alpha }_2{a}_{\mathrm{i}+1}-{\displaystyle \frac{\mathrm{i}!}{\left(\mathrm{i}-1\right)!}}{\alpha }_1{a}_{\mathrm{i}}+{\alpha }_0{a}_{\mathrm{i}-1}=0 \end{gathered}$$

(D7)

where the parameters ${\alpha }_0$ to ${\alpha }_7$ are given by Equation (D5b) in Appendix D.

2.6. Solutions of ${\Omega }^0$ on the Rigid and Free Surfaces ${\Omega }_r^0$ and ${\Omega }_f^0$

Though Equation (D7) has an infinite number of coefficient $a_{\mathrm{i}}$, as will be shown, the boundary conditions require only three coefficients, i.e., $a_0$, $a_1,$ and $a_2$. To derive the exact value of each $a_{\mathrm{i}}$, the contribution of the higher-order coefficients must be considered. In the present case, however, in view of the series converging to zero, we make an approximation that the higher order contribution i = 2 is represented by $a_2$ itself.

In accordance with such a scheme, putting $a_{\mathrm{i}}=0$ for $\mathrm{i}>2$ in Equation (D7), we get the simplified relationship between the coefficients of the general solution of ${\mathsf{\Omega }}^0$.

$$2{\alpha }_2{a}_2-{\alpha }_1{a}_1+{\alpha }_0{a}_0=0$$

(18a)

where from Equation (D5b),

$${\alpha }_0={Q^{*}}^2{k}^4$$

(18b)

$${\alpha }_1=2k^3\left({T^{*}}^2{{+2Q}^{*}}^2\right)$$

(18c)

$${\alpha }_2=k^2\left(5{T^{*}}^2-8Q^{*}{k}^2-2{Q^{*}}^2\right)$$

(18d)

Under the condition Equation (18a), from Equation (D1), ${\mathsf{\Omega }}^0$ should be solved as

$${\mathsf{\Omega }}^0=\left(a_0+a_{1}z+a_2{z}^2\right){\mathrm{e}}^{-kz}$$

(19)

2.6.1. Amplitude of the Vorticity Fluctuation on the Rigid Surfaces

The vorticity induced by the system rotation and magnetic field on the rigid surfaces follows the boundary condition Equation (17a). To derive the expression of ${\mathsf{\Omega }}^0$ on the rigid surface, substituting Equation (19) into Equation (17a), we first obtain

$$a_0=0$$

(20)

Then, inserting Equation (20) into Equation (18a), we get the following relationship.

$$k{a}_1={\gamma }_{\mathrm{r}}^{*}{a}_2$$

(21)

where ${\gamma }_{\mathrm{r}}^{*}$ is the vorticity coefficient of the rigid surface.

$${\gamma }_{\mathrm{r}}^{*}\equiv {\displaystyle \frac{5{T^{*}}^2-2{Q^{*}}^2-8Q^{*}{k}^2}{{T^{*}}^2+2{Q^{*}}^2}}$$

(22)

where subscript ‘r’ implies the rigid surface. Substitution of Equations (20) and (21) into Equation (19) leads to the amplitude of the vorticity fluctuation on the rigid surfaces.

$${\mathsf{\Omega }}_{\mathrm{r}}^0=\left({\displaystyle \frac{{\gamma }_{\mathrm{r}}^{*}}{k}}z+z^2\right)a_2{\mathrm{e}}^{-kz}$$

(23)

2.6.2. Amplitude of the Vorticity Fluctuation on the Free Surfaces

The vorticity on the free surfaces follows the boundary condition Equation (17b). Substituting Equation (19) into Equation (17b), we have

$$a_1=k{a}_0$$

(24)

Replacing $a_1$ from Equation (18a) with $a_1$ in Equation (24), we get

$$k^2{a}_0={\gamma }_{\mathrm{f}}^{*}{a}_2$$

(25)

where ${\gamma }_{\mathrm{f}}^{*}$ is the vorticity coefficient of the free surface.

$${\gamma }_{\mathrm{f}}^{*}\equiv {\displaystyle \frac{2\left(5{T^{*}}^2-2{Q^{*}}^2-8Q^{*}{k}^2\right)}{2{T^{*}}^2+3{Q^{*}}^2}}$$

(26)

where subscript ‘f’ implies the free surface. Substituting Equations (24) and (25) into Equation (19), we obtain the amplitude of the vorticity on the free surfaces.

$${\mathsf{\Omega }}_{\mathrm{f}}^0=\left({\displaystyle \frac{1}{k}}+z+{{\displaystyle \frac{k}{{\gamma }_{\mathrm{f}}^{*}}}z}^2\right)a_1{\mathrm{e}}^{-kz}$$

(27)

2.7. General and Special Solutions of the Amplitude of the Velocity Fluctuation $W^0$

In accordance with the usual differential equation theory, $W^0$ in Equation (C5) is decomposed to the general solution $W_{\mathrm{g}}^0$ and the special solution $W_{\mathrm{s}}^0$ so that they will be separately solved in the following.

2.7.1. The General Solution $W_{\mathrm{g}}^0$

$W_{\mathrm{g}}^0$ satisfies the following equation.

$$\left\{{\left({\mathrm{D}}^2-k^2\right)}^2-Q^{*}{\mathrm{D}}^2\right\}{W}_{\mathrm{g}}^0=0$$

(28)

In the same way as for ${\mathsf{\Omega }}^0$ in Equation (D1), the function form is assumed as

$$W_{\mathrm{g}}^0={\mathrm{g}}_{\mathrm{g}}\left(z,t\right){\mathrm{e}}^{-kz}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} 0\le kz\ll 1$$

(29)

and ${\mathrm{g}}_{\mathrm{g}}\left(z,t\right)$ is expressed as

$${\mathrm{g}}_{\mathrm{g}}\left(z,t\right)=A_0^{\mathrm{g}}+A_1^{\mathrm{g}}z+A_2^{\mathrm{g}}{z}^2+\cdots \cdots \cdots ={\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}{A}_{\mathrm{i}-1}^{\mathrm{g}}{z}^{\mathrm{i}-1}$$

(30)

where $A_0^{\mathrm{g}}$, $A_1^{\mathrm{g}}$, $A_2^{\mathrm{g}}$, $\cdots \cdots \cdots$ are the arbitrary coefficients of the general solution, which are functions of $t$.

With Equations (29) and (30), in accordance with Equation (D3b), Equation (28) is expanded by the operator D.

$$\left[{\mathrm{D}}^4-4k{\mathrm{D}}^3-\left(Q^{*}-4k^2\right){\mathrm{D}}^2+2Q^{*}k\mathrm{D}-Q^{*}{k}^2\right]{\mathrm{g}}_{\mathrm{g}}\left(z,t\right)=0$$

(31)

Then, from Equation (D3a), the following equation is derived.

$${\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}\left[{\displaystyle \frac{\left(\mathrm{i}+3\right)!}{\left(\mathrm{i}-1\right)!}}{A}_{\mathrm{i}+3}^{\mathrm{g}}\right.-4{\displaystyle \frac{\left(\mathrm{i}+2\right)!}{\left(\mathrm{i}-1\right)!}}k{A}_{\mathrm{i}+2}^{\mathrm{g}}-{\displaystyle \frac{\left(\mathrm{i}+1\right)!}{\left(\mathrm{i}-1\right)!}}\left(Q^{*}-4k^2\right)A_{\mathrm{i}+1}^{\mathrm{g}}+2{\displaystyle \frac{\mathrm{i}!}{\left(\mathrm{i}-1\right)!}}{Q}^{*}k{A}_{\mathrm{i}}^{\mathrm{g}}-\left.Q^{*}{k}^2{A}_{\mathrm{i}-1}^{\mathrm{g}}\right]z^{\mathrm{i}-1}=0$$

(32)

Though Equation (32) has infinite number of coefficients, as will be shown later, the boundary conditions require only three coefficients $A_0^{\mathrm{g}}$, $A_1^{\mathrm{g}}$, and $A_2^{\mathrm{g}}$. However, to derive the exact value of $A_{\mathrm{i}}^{\mathrm{g}}$, the contribution of higher order coefficients $A_{\mathrm{i}+1}^{\mathrm{g}}$, $A_{\mathrm{i}+2}^{\mathrm{g}}$, $\cdots \cdots \cdots$ than $A_2^{\mathrm{g}}$ must be considered. Considering that they form a series converging to zero, we make an approximation that the higher order coefficients than $A_2^{\mathrm{g}}$ is represented by $A_3^{\mathrm{g}}$. Due to the identity concerning z, each term of $z^{\mathrm{i}-1}$ in Equation (32) is equalized to zero.

For $\mathrm{i}=1,$ we have

$$-24k{A}_3^{\mathrm{g}}-2\left(Q^{*}-4k^2\right)A_2^{\mathrm{g}}+2Q^{*}k{A}_1^{\mathrm{g}}-Q^{*}{k}^2{A}_0^{\mathrm{g}}=0$$

(33a)

For $\mathrm{i}=2$, we get

$$-6\left(Q^{*}-4k^2\right)A_3^{\mathrm{g}}+4Q^{*}k{A}_2^{\mathrm{g}}-Q^{*}{k}^2{A}_1^{\mathrm{g}}=0$$

(33b)

From Equations (33a) and (33b), the terms of $A_3^{\mathrm{g}}$ are eliminated, so that we finally obtain the equation of $A_0^{\mathrm{g}}$, $A_1^{\mathrm{g}},$ and $A_2^{\mathrm{g}}$.

$$-2\left(Q^{*}+16k^4\right)A_2^{\mathrm{g}}+2Q^{*}k{\left(Q^{*}-2k^2\right)A}_1^{\mathrm{g}}-Q^{*}{k}^2\left(Q^{*}-4k^2\right)A_0^{\mathrm{g}}=0$$

(34)

2.7.2. The Special Solution $W_{\mathrm{s}}^0$

As for the special solution $W_{\mathrm{s}}^0$, Equation (C5) leads to the equation.

$$\left\{{\left({\mathrm{D}}^2-k^2\right)}^2-Q^{*}{\mathrm{D}}^2\right\}{W}_{\mathrm{s}}^0=T^{*}\mathrm{D}{\mathsf{\Omega }}^0$$

(35)

According to Equation (29), $W_{\mathrm{s}}^0$ is also written by

$$W_{\mathrm{s}}^0={\mathrm{g}}_{\mathrm{s}}\left(z,t\right){\mathrm{e}}^{-kz}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} 0\le kz\ll 1$$

(36)

where ${\mathrm{g}}_{\mathrm{s}}\left(z,t\right)$ is expressed as

$${\mathrm{g}}_{\mathrm{s}}\left(z,t\right)=A_0^{\mathrm{s}}+A_1^{\mathrm{s}}z+A_2^{\mathrm{s}}{z}^2+\cdots \cdots \cdots ={\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}{A}_{\mathrm{i}-1}^{\mathrm{s}}{z}^{\mathrm{i}-1}$$

(37)

where $A_0^{\mathrm{s}}$, $A_1^{\mathrm{s}}$, $A_2^{\mathrm{s}}$, $\cdots \cdots \cdots$ are the arbitrary coefficients of the special solution, which are functions of $t$.

From Equations (D3a) and (D3b), we have the right-hand side of Equation (35) as a series of z.

$$\mathrm{D}{\mathsf{\Omega }}^0={\mathrm{e}}^{-kz}{\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}\left[{\displaystyle \frac{\mathrm{i}!}{\left(\mathrm{i}-1\right)!}}{a}_{\mathrm{i}}-k{a}_{\mathrm{i}-1}\right]z^{\mathrm{i}-1}$$

(38)

Using Equations (D3a), (D3b) and (36), we can expand the left-hand side of Equation (35) with regard to z.

$$\begin{array}{l}\left\{{\left({\mathrm{D}}^2-k^2\right)}^2-Q^{*}{\mathrm{D}}^2\right\}{W}_{\mathrm{s}}^0\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\mathrm{e}}^{-kz}{\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}\left[{\displaystyle \frac{\left(\mathrm{i}+3\right)!}{\left(\mathrm{i}-1\right)!}}{A}_{\mathrm{i}+3}^s\right.-4{\displaystyle \frac{\left(\mathrm{i}+2\right)!}{\left(\mathrm{i}-1\right)!}}k{A}_{\mathrm{i}+2}^{\mathrm{s}}-{\displaystyle \frac{\left(\mathrm{i}+1\right)!}{\left(\mathrm{i}-1\right)!}}\left(Q^{*}-4k^2\right)A_{\mathrm{i}+1}^{\mathrm{s}}+2{\displaystyle \frac{\mathrm{i}!}{\left(\mathrm{i}-1\right)!}}{Q}^{*}k{A}_{\mathrm{i}}^{\mathrm{s}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left.Q^{*}{k}^2{A}_{\mathrm{i}-1}^{\mathrm{s}}\right]z^{\mathrm{i}-1}\end{array}$$

(39)

Substituting Equations (38) and (39) into Equation (35), and comparing each term with the same power of z, we can pick up the following equations.

For $\mathrm{i}=1:$

$$-2\left(Q^{*}-4k^2\right)A_2^{\mathrm{s}}+2Q^{*}k{A}_1^{\mathrm{s}}-Q^{*}{k}^2{A}_0^{\mathrm{s}}=T^{*}\left(a_1-k{a}_0\right)$$

(40a)

For $\mathrm{i}=2:$

$$4Q^{*}k{A}_2^{\mathrm{s}}-Q^{*}{k}^2{A}_1^{\mathrm{s}}=T^{*}\left(2a_2-k{a}_1\right)$$

(40b)

For $\mathrm{i}=3:$

$$Q^{*}k{A}_2^{\mathrm{s}}=T^{*}{a}_2$$

(40c)

Therefore, from Equation (40c), the coefficient $A_2^{\mathrm{s}}$ is given by $a_2.$

$$A_2^{\mathrm{s}}={\displaystyle \frac{T^{*}}{Q^{*}k}}{a}_2$$

(41a)

Substituting Equation (41a) into Equation (40b), $A_1^{\mathrm{s}}$ is expressed by $a_1$ and $a_2.$

$$A_1^{\mathrm{s}}={\displaystyle \frac{T^{*}}{Q^{*}{k}^2}}\left(2a_2+k{a}_1\right)$$

(41b)

Inserting Equations (41a) and (41b) into Equation (40a), $A_0^{\mathrm{s}}$ is represented by $a_0$, $a_1,$ and $a_2.$

$$A_0^{\mathrm{s}}={\displaystyle \frac{T^{*}}{{Q^{*}}^2{k}^3}}\left\{2\left({4k}^2+Q^{*}\right)a_2{+Q}^{*}k{a}_1+Q^{*}{k}^2{a}_0\right\}$$

(41c)

2.8. Solutions of the Rigid- and Free-Surface Amplitudes $W_r^0$ and $W_f^0$

The solution of $W^0$ is expressed as

$$W^0=\left(A_0+A_{1}z+A_2{z}^2\right){\mathrm{e}}^{-kz}$$

(42)

Then, from Equations (29) and (30), the general solution is provided by

$$W_{\mathrm{g}}^0=\left(A_0^{\mathrm{g}}+A_1^{\mathrm{g}}z+A_2^{\mathrm{g}}{z}^2\right){\mathrm{e}}^{-kz}$$

(43)

with the coefficient equation, Equation (34).

On the other hand, from Equations (36) and (37), the special solution is written by

$$W_{\mathrm{s}}^0=\left(A_0^{\mathrm{s}}+A_1^{\mathrm{s}}z+A_2^{\mathrm{s}}{z}^2\right){\mathrm{e}}^{-kz}$$

(44)

Since the solution of $W^0$ is composed of the sum of the general solution $W_{\mathrm{g}}^0$ and the special solution $W_{\mathrm{s}}^0,$ we have the following relationship between their coefficients.

$$A_{\mathrm{i}}=A_{\mathrm{i}}^{\mathrm{g}}+A_{\mathrm{i}}^{\mathrm{s}}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{i} = 0 \mathrm{t}\mathrm{o} 2$$

(45)

Substituting Equation (45) into Equation (34), we derive the following equation.

$$-2\left(Q^{*}+16k^4\right)\left(A_2-A_2^{\mathrm{s}}\right)+2Q^{*}k\left(Q^{*}-2k^2\right)\left(A_1-A_1^{\mathrm{s}}\right)-Q^{*}{k}^2\left(Q^{*}-4k^2\right)\left(A_0-A_0^{\mathrm{s}}\right)=0$$

(46)

Under the condition of Equation (46), the solutions on the rigid and free surfaces are derived.

2.8.1. The Rigid-Surface Solution $W_{\mathrm{r}}^0$

From the general boundary condition for a solid surface, $W^0=0$ for z $=0$ in Equation (15), we have

$$A_0=0$$

(47)

For the rigid surface, the boundary condition $\mathrm{D}{W}^0=0$ for z $=0$ in Equation (17a) is fulfilled, so that substituting Equation (42) into Equation (17a) and inserting Equation (47) into the resulting equation, we obtain

$$A_1=0$$

(48)

Here, the rigid-surface conditions of the vorticity were derived by $a_0=0$ in Equation (20) and $k{a}_1={\gamma }_{\mathrm{r}}^{*}{a}_2$ in Equation (21). Equations (47) and (48) are first substituted into Equation (46), then together with Equations (20) and (21), Equations (41a)–(41c) are inserted into the resulting equation, so that we finally get

$$A_2={\displaystyle \frac{T^{*}\left(8k^2-Q^{*}{\gamma }_{\mathrm{r}}^{*}\right)}{2k\left(16k^4+{Q^{*}}^2\right)}}{a}_2$$

(49a)

To describe the equation in Equation (49a) more explicitly, under the electrochemical condition $k^2\gg Q^{*}$, using Equation (22), we rewrite Equation (49a) as follows.

$$A_2={\displaystyle \frac{T^{*}\left({T^{*}}^2+3{Q^{*}}^2\right)}{4k^3\left({T^{*}}^2+{Q^{*}}^2\right)}}{a}_2$$

(49b)

The relationship $k^2\gg Q^{*}$ is generally obtained in any electrochemical system from the low electric conductivity of electrolyte solution $\left({\sigma }^{*}~10 \mathrm{S}{\mathrm{m}}^2\right)$ and the scale of length of nucleation $\left(d<{10}^{-4} \mathrm{m}\right)$, i.e., $k^2>{10}^8 {\mathrm{m}}^{-2}$ and $Q^{*}<$ ${10}^4$ ${\mathrm{m}}^{-2}$ are obtained.

In view of Equations (47) and (48), $W_{\mathrm{r}}^0$ is expressed as

$$W_{\mathrm{r}}^0=A_2{z}^2{\mathrm{e}}^{-kz}$$

(50)

2.8.2. The Free-Surface Solution $W_{\mathrm{f}}^0$

In addition to the condition $A_0=0$ in Equation (47), from the free-surface condition ${\mathrm{D}}^2{\mathrm{W}}^0=0$ for z $=0$ in Equation (17b), we have

$$A_2={kA}_1$$

(51)

As for the coefficients of the vorticity, it follows that $a_1=k{a}_0$ in Equation (24) and $k^2{a}_0={\gamma }_{\mathrm{f}}^{*}{a}_2$ in Equation (25) are derived for the free-surface case. Using Equations (47) and (51) together with Equations (24) and (25), in the same way as Equation (49a) in the rigid-surface case, we obtain

$$A_1=-{\displaystyle \frac{T^{*}\left({\gamma }_{\mathrm{f}}^{*}-2\right)}{k^2\left(8k^4+Q^{*}\right)}}{a}_1$$

(52a)

Furthermore, under the electrochemical condition $k^2\gg Q^{*}$, with Equation (26), we rewrite Equation (52a) as follows.

$$A_1=-{\displaystyle \frac{T^{*}\left(3{T^{*}}^2-8Q^{*}{k}^2\right)}{4k^4\left({{2T}^{*}}^2+{Q^{*}}^2\right)}}{a}_1$$

(52b)

Finally, in view of Equations (47) and (51), $W_{\mathrm{f}}^0$ is written as

$$W_{\mathrm{f}}^0=A_1\left(1+kz\right)z{\mathrm{e}}^{-kz}$$

(53)

2.9. Solutions of the Amplitudes of the Rigid- and Free-Surface Concentration Fluctuations ${\Theta }_r^0$ and ${\Theta }_f^0$

The solution ${\mathsf{\Theta }}^0$ of the concentration fluctuation amplitude is also described by the sum of the general solution ${\mathsf{\Theta }}_{\mathrm{g}}^0$ and the special solution ${\mathsf{\Theta }}_{\mathrm{s}}^0$.

From Equation (C3e), the general solution satisfies

$$\left({\mathrm{D}}^2-k^2\right){\mathsf{\Theta }}_{\mathrm{g}}^0=0$$

(54)

Applying the boundary condition Equation (14),

$${\mathsf{\Theta }}_{\mathrm{g}}^0=0\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} z=\mathrm{\infty }$$

(55)

we get

$${\mathsf{\Theta }}_{\mathrm{g}}^0=B_1{\mathrm{e}}^{-kz}$$

(56)

where $B_1$ is an arbitrary constant as a function of t.

Then, the special solution is differently solved for the rigid and free surfaces.

$$\left({\mathrm{D}}^2-k^2\right){\mathsf{\Theta }}_{\mathrm{i}\mathrm{s}}^0=R^{*}{W}_{\mathrm{i}}^0\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{i} = \mathrm{r} \mathrm{o}\mathrm{r} \mathrm{f}$$

(57)

where $W_{\mathrm{r}}^0$ and $W_{\mathrm{f}}^0$ are the rigid- and free-surface velocity amplitudes in Equations (50) and (53), respectively. $R^{*}$ is the mass transfer coefficient (mol s m⁻⁶) defined by the average concentration gradient $L_{\mathrm{m}}$ in the diffusion layer and the diffusion coefficient $D_{\mathrm{m}}$.

$$R^{*}\equiv {\displaystyle \frac{L_{\mathrm{m}}}{D_{\mathrm{m}}}}$$

(58)

$R^{*}$ changes the sign with $L_{\mathrm{m}}$, which is positive for a cathodic reaction, and negative for an anodic reaction.

2.9.1. The Rigid-Surface Solution ${\mathsf{\Theta }}_{\mathrm{r}}^0$

Equation (57) is solved for the rigid-surface as follows.

$${\mathsf{\Theta }}_{\mathrm{r}\mathrm{s}}^0=-{\displaystyle \frac{A_2{R}^{*}}{24k^4}}\left(4k^3{z}^3+6k^2{z}^2+6kz+3\right)e^{-kz}$$

(59)

The solution of the rigid-surface ${\mathsf{\Theta }}_{\mathrm{r}}^0$ is therefore expressed by the sum of Equations (56) and (59).

$${\mathsf{\Theta }}_{\mathrm{r}}^0{=B}_{\mathrm{r}1}{\mathrm{e}}^{-kz}-{\displaystyle \frac{A_2{R}^{*}}{24k^4}}\left(4k^3{z}^3+6k^2{z}^2+6kz+3\right)e^{-kz}$$

(60a)

The derivative is given by

$$\mathrm{D}{\mathsf{\Theta }}_{\mathrm{r}}^0{=-kB}_{\mathrm{r}1}{\mathrm{e}}^{-kz}-{\displaystyle \frac{A_2{R}^{*}}{24k^3}}\left(-4k^3{z}^3+6k^2{z}^2+3\right)e^{-kz}$$

(60b)

2.9.2. The Free-Surface Solution ${\mathsf{\Theta }}_{\mathrm{f}}^0$

Equation (57) is solved for the free-surface case as follows.

$${\mathsf{\Theta }}_{\mathrm{f}\mathrm{s}}^0=-{\displaystyle \frac{A_1{R}^{*}}{12k^3}}\left(2k^3{z}^3+6k^2{z}^2+6kz+3\right)e^{-kz}$$

(61)

The solution of the free-surface amplitude ${\mathsf{\Theta }}_{\mathrm{f}}^0$ is written by the sum of Equations (56) and (61).

$${\mathsf{\Theta }}_{\mathrm{f}}^0{=B}_{\mathrm{f}1}{\mathrm{e}}^{-kz}-{\displaystyle \frac{A_2{R}^{*}}{12k^3}}\left(2k^3{z}^3+6k^2{z}^2+6kz+3\right)e^{-kz}$$

(62a)

The derivative is

$$\mathrm{D}{\mathsf{\Theta }}_{\mathrm{f}}^0{=-kB}_{\mathrm{f}1}{\mathrm{e}}^{-kz}-{\displaystyle \frac{A_1{R}^{*}}{12k^2}}\left(-2k^3{z}^3+6kz+3\right)e^{-kz}$$

(62b)

2.10. Vortex Filter Functions on the Rigid and Free Surfaces $f_r\left(k\right)$ and $f_f\left(k\right)$

As has been shown in the previous papers [16,17], the nonequilibrium fluctuations in magnetoelectrodeposition (MED) under a VMHDF are described by their amplitude factors, and according to the amplitude factor, they exponentially develop with time in the unstable case, and dwindle in the stable case. An amplitude factor is composed of the product of two parts; one is a vortex filter function, where wavenumber components necessary for the electrode reaction are selected from the solution-side fluctuations through MHD vortex rotations, whereas the other corresponds to the electrode reaction process, such as electrodeposition, utilizing the selected components. As a result, if the vortex motions were prohibited, the amplitude factor would converge to the one in a static solution without a magnetic field and system rotation. This means that to describe the vortex behaviors in the solution, it is sufficient to examine the vortex filter function. In the following, for examining the role of the chiral vortex motions in magnetoelectrochemical reactions in a rotating system, we pick up the filter function.

Whether rigid or free, from Equation (C3c), the following relationship among the amplitudes of the vorticity ${\mathsf{\Omega }}^0$, velocity $W^0,$ and current density $J^0$ is derived.

$$S^{*}\mathrm{D}{J}^0=-\left({\mathrm{D}}^2-k^2\right){\mathsf{\Omega }}^0-T^{*}\mathrm{D}{W}^0$$

(63a)

where $T^{*}$ is defined by Equation (C4c), and $S^{*}$ is the MHD coefficient (T m s kg⁻¹) expressed by

$$S^{*}\equiv {\displaystyle \frac{B_0}{\rho \mathsf{\nu }}}$$

(63b)

In view of the fact that the fluctuations vanish at a distance far away from the surface together with Equation (15), we have

$$W^0=0\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} z=0 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{\infty }$$

(64a)

$$J^0=0\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} z=\mathrm{\infty }$$

(64b)

To deduce the amplitude $J^0$ for $z=0,$ i.e., $J^0\left(0,t\right)$ of the current density fluctuation on the rigid and free surfaces, Equation (63a) is integrated with regard to z from $\mathrm{z}=0$ to $\infty$. Using the boundary conditions Equations (64a) and (64b), we obtain the integrated equation

$$S^{*}{J}^0\left(0,t\right)={\int }_0^{\mathrm{\infty }}\left({\mathrm{D}}^2-k^2\right){\mathsf{\Omega }}^0\mathrm{d}z$$

(65)

Since using the formulas on ${\mathsf{\Omega }}^0$ in Equations (D3a) and (D3b), we obtain the equation of $\left({\mathrm{D}}^2-k^2\right){\mathsf{\Omega }}^0$

$$\left({\mathrm{D}}^2-k^2\right){\mathsf{\Omega }}^0={\mathrm{e}}^{-kz}\left\{2\left(a_2-k{a}_1\right)-4k{a}_{2}z\right\}$$

(66)

we can calculate the following integration.

$${\int }_0^{\mathrm{\infty }}\left({\mathrm{D}}^2-k^2\right){\mathsf{\Omega }}^0\mathrm{d}z=-{\displaystyle \frac{2}{k}}\left(a_2+k{a}_1\right)$$

(67)

Substituting Equation (67) into Equation (65), we get

$$J^0\left(0,t\right)=-{\displaystyle \frac{2}{S^{*}k}}\left(a_2+k{a}_1\right)$$

(68)

From the boundary condition of the current density fluctuation Equation (16), we have

$$\mathrm{D}{\mathsf{\Theta }}^0\left(0,t\right)=-{\displaystyle \frac{J^0\left(0,t\right)}{z_{\mathrm{m}}F{D}_{\mathrm{m}}}}$$

(69)

Substituting for $J^0\left(0,t\right)$ from Equation (68) in Equation (69), we obtain the boundary condition of $\mathrm{D}{\mathsf{\Theta }}^0$ related with the vorticity, which is separately solved for the rigid- and free-surface cases.

$$\mathrm{D}{\mathsf{\Theta }}^0\left(0,t\right)={\displaystyle \frac{2\left(a_2+k{a}_1\right)}{z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}k}}$$

(70)

2.10.1. Rigid-Surface Vortex (RV) Filter Function

Inserting Equation (21) into Equation (70), and using the resultant equation and Equations (22) and (49b), we rewrite Equation (60b) at the rigid surface as a rigid-surface boundary condition.

$$\mathrm{D}{\mathsf{\Theta }}_{\mathrm{r}}^0\left(0,t\right)={\displaystyle \frac{4\left\{3-4{\left(k/\epsilon \right)}^2\right\}}{\left(1+2{\xi }^2\right)z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}k}}{a}_2$$

(71a)

where

$$\epsilon \equiv {\left({\displaystyle \frac{T^{*2}}{Q^{*}}}\right)}^{1/2}\left[={\left\{{\displaystyle \frac{4\rho }{{\sigma }^{*}\nu }}{\left({\displaystyle \frac{\mathsf{\Omega }}{B_0}}\right)}^2\right\}}^{1/2}\right]$$

(71b)

and

$$\xi \equiv {\displaystyle \frac{Q^{*}}{T^{*}}}\left[={\displaystyle \frac{{\sigma }^{*}{B_0}^2}{2\rho \mathsf{\Omega }}}\right]$$

(71c)

Here, for simplicity, assuming the electrochemical condition $k^2\gg Q^{*},$ from Equations (49b), (60a), (60b) and (71a), we get another rigid-surface boundary equation, i.e.,

$${\mathsf{\Theta }}_{\mathrm{r}}^0\left(0,t\right)=-{\displaystyle \frac{64k^5\left\{3-4{\left(k/\epsilon \right)}^2\right\}+\left(1+3{\xi }^2\right)z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}{T}^{*}{R}^{*}}{16k^7\left(1+2{\xi }^2\right)z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}}}{a}_2$$

(72)

To obtain the RV filter function ${\mathrm{f}}_{\mathrm{r}}\left(k\right)$ selecting proper fluctuation components through the rigid-surface vortices, we divide Equation (71a) by Equation (72), obtaining $\mathrm{D}{\mathsf{\Theta }}_{\mathrm{r}}^0\left(0,t\right)/{\mathsf{\Theta }}_{\mathrm{r}}^0\left(0,t\right)$, which is expressed by the following fractional expression of the denominator function ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ and the numerator function ${\mathrm{g}}_{\mathrm{r}}\left(k\right)$ [16,17].

$${\mathrm{f}}_{\mathrm{r}}\left(k\right)\equiv -{\displaystyle \frac{{\mathrm{g}}_{\mathrm{r}}\left(k\right)}{{\mathrm{h}}_{\mathrm{r}}\left(k\right)}}$$

(73a)

where ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{g}}_{\mathrm{r}}\left(k\right)$ are expressed as

$${\mathrm{h}}_{\mathrm{r}}\left(k\right)=64k^5\left\{3-4{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}+\left(1+3{\xi }^2\right)z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}{T}^{*}{R}^{*}$$

(73b)

and

$${\mathrm{g}}_{\mathrm{r}}\left(k\right)=64k^6\left\{3-4{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}$$

(73c)

2.10.2. Free-Surface Vortex (FV) Filter Function

Inserting Equations (24) and (25) into Equation (70), and using the resultant equation with Equations (26) and (52b) under the electrochemical condition $k^2\gg {\mathrm{Q}}^{*}$, from Equation (62b), we can derive a free-surface boundary condition.

$$\mathrm{D}{\mathsf{\Theta }}_{\mathrm{f}}^0\left(0,t\right)={\displaystyle \frac{4\left\{3-4{\left(k/\epsilon \right)}^2\right\}}{\left\{5-8{\left(k/\epsilon \right)}^2\right\}{z}_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}}}{a}_1$$

(74a)

Then, using the same electrochemical condition $k^2\gg Q^{*}$ for Equation (26), we can rewrite Equation (62a) as another free-surface boundary condition.

$${\mathsf{\Theta }}_{\mathrm{f}}^0\left(0,t\right)=-{\displaystyle \frac{32k^6\left(2+{3\xi }^2\right)\left\{3-4{\left(k/\epsilon \right)}^2\right\}-z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}{T}^{*}{R}^{*}\left\{3-8{\left(k/\epsilon \right)}^2\right\}\left\{5-8{\left(k/\epsilon \right)}^2\right\}}{32k^7\left(2+{\xi }^2\right)\left\{5-8{\left(k/\epsilon \right)}^2\right\}{z}_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}}}{a}_1$$

(74b)

The FV filter function of the rigid-surface vortexes is thus defined by

$${\mathrm{f}}_{\mathrm{f}}\left(k\right)\equiv -{\displaystyle \frac{{\mathrm{g}}_{\mathrm{f}}\left(k\right)}{{\mathrm{h}}_{\mathrm{f}}\left(k\right)}}$$

(75a)

where ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$ and ${\mathrm{g}}_{\mathrm{f}}\left(k\right)$ are expressed as

$${\mathrm{h}}_{\mathrm{f}}\left(k\right)=32k^6\left(2+{3\xi }^2\right)\left\{3-4{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}-z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}{T}^{*}{R}^{*}\left\{3-8{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}\left\{5-8{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}$$

(75b)

and

$${\mathrm{g}}_{\mathrm{f}}\left(k\right)=32k^7\left(2+{\xi }^2\right)\left\{3-4{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}$$

(75c)

When the system rotation is stopped, $\mathsf{\Omega }=0$, i.e., $T^{*}=0$ is inserted into Equations (73a) and (75a), so that we get the filter functions ${\mathrm{f}}_{\mathrm{r}}\left(k\right)={\mathrm{f}}_{\mathrm{f}}\left(k\right)=-k$, which are equal to those of the stationary solution [18].

2.11. Precessional Conditions of the Microscopic RMHD Vortices

To distinguish MHD vortices in a rotating system from those in a stationary system, we especially call the former vortices as rotational MHD (RMHD) vortices. For an RMHD vortex to contribute to a chiral reaction process such as chiral nucleation, some interaction between the vortex and the mass flux of the reagents is required. However, due to the nature of axial vector, the z-component of the vorticity vector induced by a vertical magnetic field rotates horizontally along the x-y plane, and therefore does not make interplay with the mass flux vector having only z-component (see Figure 6). The system rotation bestows precessional motion on the vorticity vectors to interact with the mass flux vector. This means that to assess the mechanism correctly, the roles of not only the vorticity itself, but also the system rotation, must be examined.

A new quantity describing the precessional motions, i.e., Coriolis vorticity ${\tilde{\omega }}_{\mathrm{z}}$(m⁻² s⁻¹) is thus introduced.

$${\tilde{\omega }}_z\equiv T^{*}{\omega }_z$$

(76)

where $T^{*}$ is a scalar quantity shown in Equation (C4c), so that ${\tilde{\omega }}_z$ is the same axial vector as ${\omega }_z$. The rotational direction of an RMHD vortex is not determined by the sign of ${\omega }_z$, but by the sign of ${\tilde{\omega }}_z.$

Using the Coriolis vorticity, we can rewrite the z-component of the velocity in Equation (B17d).

$${\displaystyle \frac{1}{\mathsf{\nu }}}{\displaystyle \frac{\partial }{\partial t}}{\nabla }^{2}w={\nabla }^{4}w+{\displaystyle \frac{B_0}{\rho \mathsf{\nu }{\mu }_0}}{\displaystyle \frac{\partial }{\partial z}}{\nabla }^2{b}_{\mathrm{z}}-{\displaystyle \frac{\partial }{\partial z}}{\tilde{\omega }}_z$$

(77)

As shown in Equation (77), the z-component of the velocity, $w$ receives precessional motion by ${\tilde{\omega }}_{\mathrm{z}}$, controlling the mass transfer for chiral deposition. In a stationary system, as has been discussed in the preceding papers [18,19], a precessional motion transfers from the 1st generation to the 3rd generation indirectly through MHD vortex rotations in each generation. However, in the present case, it is directly donated to all the vortices by the system rotation without any contribution of electrochemical reactions.

In the same way as ordinary vorticity ${\omega }_z$, the Coriolis vorticity determines the rotational directions of RMHD vortices; the positive and negative values of ${\tilde{\omega }}_{\mathrm{z}}$ indicate anticlockwise (ACW) and clockwise (CW) rotations, respectively. To derive a more explicit form of ${\tilde{\omega }}_z$, Equation (76) is transformed to the following amplitude equation by using the Fourier transform with respect to the x- and y-coordinates.

$${\tilde{\mathsf{\Omega }}}^0\left(z,t\right)=T^{*}{\mathsf{\Omega }}^0\left(z,t\right)$$

(78)

where ${\tilde{\mathsf{\Omega }}}^0$ is the amplitude of the Coriolis vorticity ${\tilde{\omega }}_{\mathrm{z}}$ (m⁻² s⁻¹).

Substituting Equation (16) into Equations (71a) and (74a), and inserting the resultant equations into Equations (23) and (27), from Equation (78), we obtain the rigid-surface and free-surface amplitudes of the Coriolis vorticity.

$${\tilde{\mathsf{\Omega }}}_{\mathrm{r}}^0\left(z,t\right)=-S^{*}{T}^{*}{\tilde{J}}_{\mathrm{r}}^0\left(z,t\right)$$

(79a)

and

$${\tilde{\mathsf{\Omega }}}_{\mathrm{f}}^0\left(z,t\right)=-S^{*}{T}^{*}{\tilde{J}}_{\mathrm{f}}^0\left(z,t\right)$$

(79b)

where ${\tilde{J}}_{\mathrm{r}}^0\left(z,t\right)$ and ${\tilde{J}}_{\mathrm{f}}^0\left(z,t\right)$ are the amplitudes of the modified current density fluctuations (A m⁻¹) defined as

$${\tilde{J}}_{\mathrm{r}}^0\left(z,t\right)\equiv J_{\mathrm{r}}^0\left(0,t\right){\displaystyle \frac{1+2{\xi }^2}{4\left\{3-4{\left(k/\epsilon \right)}^2\right\}}}\left({\gamma }_{\mathrm{r}}^{*}z+{kz}^2\right)e^{-kz}$$

(80a)

and

$${\tilde{J}}_{\mathrm{f}}^0\left(z,t\right)\equiv J_{\mathrm{f}}^0\left(0,t\right){\displaystyle \frac{5-8{\left(k/\epsilon \right)}^2}{2\left\{3-4{\left(k/\epsilon \right)}^2\right\}}}\left({\displaystyle \frac{1}{k}}+z+{{\displaystyle \frac{k}{{\gamma }_{\mathrm{f}}^{*}}}z}^2\right)e^{-kz}$$

(80b)

Here, under the electrochemical condition $k^2\gg Q^{*},$ the vorticity coefficients of the rigid and free surfaces, ${\gamma }_{\mathrm{r}}^{*}$ and ${\gamma }_{\mathrm{f}}^{*}$ in Equations (22) and (26) are rewritten as

$${\gamma }_{\mathrm{r}}^{*}={\displaystyle \frac{5-8{\left(k/\epsilon \right)}^2}{1+2{\xi }^2}}$$

(81a)

and

$${\gamma }_{\mathrm{f}}^{*}={\displaystyle \frac{2\left\{5-8{\left(k/\epsilon \right)}^2\right\}}{2+3{\xi }^2}}$$

(81b)

The amplitudes ${\tilde{\mathsf{\Omega }}}_{\mathrm{r}}^0\left(z,t\right)$ and ${\tilde{\mathsf{\Omega }}}_{\mathrm{f}}^0\left(z,t\right)$ have therefore three singular points with regard to the wavenumber k, i.e., 0, ${\left(5/8\right)}^{1/2}\epsilon$, and ${\left(3/4\right)}^{1/2}\epsilon$ in order of size, which results from the balance between the Lorentz force and Coriolis force acting on vortices, stopping rotation. For positive wavenumber $k>0$, at $k=$ ${\left(3/4\right)}^{1/2}\epsilon$, as shown in Equations (80a) and (80b), the amplitudes of modified current density fluctuations ${\tilde{J}}_{\mathrm{r}}^0\left(z,t\right)$ and ${\tilde{J}}_{\mathrm{f}}^0\left(z,t\right)$ diverge to infinity, whereas at $k={\left(5/8\right)}^{1/2}\epsilon$, the vorticity coefficients become zero. The amplitudes have no solutions or change their signs there. This means that at the singular points, the vortices cannot exist, or change their phases by 180 degrees. Therefore, vortices containing such singular points are expected to vanish with time, leaving stationary solution areas.

Therefore, for the vortices to rotate, the following basic condition of vortex rotation, excluding such singular points, must be fulfilled.

$$0<k<k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$$

(82a)

$k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$ is the limiting singular point defined as

$${k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}\equiv \left({\displaystyle \frac{5}{8}}\right)}^{1/2}\epsilon$$

(82b)

where in view of the representative scale of k, ${Q^{*}}^{1/2}\approx 0$ is considered.

By Fourier inversion with regard to x- and y-coordinates, Equations (79a) and (79b) are transformed to the Coriolis vorticities.

$${\tilde{\omega }}_{\mathrm{z},\mathrm{j}}\left(x,y,z,t\right)=-S^{*}{T}^{*}{\tilde{j}}_{\mathrm{z},\mathrm{j}}\left(x,y,z,t\right)\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j} = \mathrm{r} \mathrm{o}\mathrm{r} \mathrm{f}$$

(83a)

where

$$S^{*}{T}^{*}={\displaystyle \frac{2B_0\mathsf{\Omega }}{\rho {\nu }^2}} \left(\propto B_0\mathsf{\Omega }\right)$$

(83b)

${\tilde{j}}_{\mathrm{z},\mathrm{j}}\left(x,y,z,t\right)$ is the modified current density fluctuation expressed as

$${\tilde{j}}_{\mathrm{z},\mathrm{j}}\left(x,y,z,t\right)={\displaystyle \frac{1}{2\pi }}{\iint }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\tilde{J}}_{\mathrm{j}}^0\left(z,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\left(k_{\mathrm{x}}x+k_{\mathrm{y}}y\right)\right]\mathrm{d}{k}_{\mathrm{x}}d{k}_{\mathrm{y}}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j} = \mathrm{r} \mathrm{o}\mathrm{r} \mathrm{f}$$

(84)

where “$\mathrm{i}$” is the unit of the imaginary number.

Under the condition of Equation (82a), the modified current fluctuation does not change the sign except for multiplying its amplitude by a minus or complex number. As a result, the phase of ${\tilde{j}}_{\mathrm{z},\mathrm{j}}\left(x,y,z,t\right)$ is equal to that of the current density fluctuation at the electrode surface.

$$j_{\mathrm{z},\mathrm{j}}\left(x,y,0,t\right)={\displaystyle \frac{1}{2\pi }}{\iint }_{-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{\mathrm{j}}^0\left(0,t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[\mathrm{i}\left(k_{\mathrm{x}}x+k_{\mathrm{y}}y\right)\right]\mathrm{d}{k}_{\mathrm{x}}\mathrm{d}{k}_{\mathrm{y}}\hspace{1em}\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j} = \mathrm{r} \mathrm{o}\mathrm{r} \mathrm{f}$$

(85)

Since the sign of the current density fluctuation is equal to that of the current density without fluctuation $J_{\mathrm{z}}^{*}$ (A m⁻²), using Equations (83a) and (83b), we can determine the sign of the Coriolis vorticity ${\tilde{\omega }}_{\mathrm{z},\mathrm{j}}$ by the sign of the product, $-{J_{\mathrm{z}}^{*}B}_0\mathsf{\Omega }$.

$$\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\left({\tilde{\omega }}_{\mathrm{z},\mathrm{j}}\right)=\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\left(-{J_{\mathrm{z}}^{*}B}_0\mathsf{\Omega }\right)$$

(86)

where Sign$\left(x\right)$ is a function representing the sign of x. Therefore, the precessional direction of RMHD vortex depends on whether the reaction is cathodic or anodic.

In case of cathodic reaction like electrodeposition, the current density is defined as negative,

$$J_{\mathrm{z}}^{*}\le 0 \text{for a cathodic reaction}$$

(87a)

so that the sign of the Coriolis vorticity ${\tilde{\omega }}_{\mathrm{z},\mathrm{j}}$, i.e., the precessional direction of the corresponding vortex flow is determined by the sign of $B_0\mathsf{\Omega }$, i.e.,

$$\mathrm{F}\mathrm{o}\mathrm{r} B_0\mathsf{\Omega }>0, {\tilde{\omega }}_{\mathrm{z},\mathrm{j}}>0 \text{(anticlockwise (ACW) rotation)}$$

(87b)

$$\mathrm{F}\mathrm{o}\mathrm{r} B_0\mathsf{\Omega }<0, {\tilde{\omega }}_{\mathrm{z},\mathrm{j}}<0 \text{(clockwise (CW) rotation)}$$

(87c)

For anodic reactions such as anodic dissolution, as the current sign is positive, we have the reverse conditions,

$$J_{\mathrm{z}}^{*}\ge 0 \text{for an anodic reaction}$$

(88a)

The sign of Coriolis vorticity ${\tilde{\omega }}_{\mathrm{z},\mathrm{j}}$, i.e., the rotational direction is reversed.

$$\mathrm{F}\mathrm{o}\mathrm{r} B_0\mathsf{\Omega }>0, {\tilde{\omega }}_{\mathrm{z},\mathrm{j}}<0 \text{(CW rotation)}$$

(88b)

$$\mathrm{F}\mathrm{o}\mathrm{r} B_0\mathsf{\Omega }<0, {\tilde{\omega }}_{\mathrm{z},\mathrm{j}}>0 \text{(ACW rotation)}$$

(88c)

2.12. Critical Condition of the Precessional Rotation

When the vortices of the three generations directly receive the precession from the system rotation, because of the largest scale of length, the 1st generation vortices will first start rotation. Triggered by the vortex motions of the 1st generation, those of the 2nd and 3rd generations will also be activated. This means that the critical condition of the precessional rotations in all the generations is consistent with that of the 1st generation.

To derive the critical condition of the precessional rotations, we therefore consider the possible area of the wavenumber $k^{\mathrm{a}}$ in the power spectrum of the asymmetrical fluctuation in the 1st generation, represented by a Gaussian-type function [18].

$${P_{\mathrm{i}\mathrm{n}\mathrm{t}}\left(k_{\mathrm{x}},k_{\mathrm{y}}\right)}^{\mathrm{a}}={\displaystyle \frac{a^{\mathrm{a}}}{\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{a^{\mathrm{a}}}^2{k}^2\right)$$

(89)

where superscript ‘a’ implies asymmetrical fluctuation in the 1st generation [18], and $a^{\mathrm{a}}$ is the autocorrelation distance (m), which is regarded as the scale of length of the 1st generation. Here, the upper limit of the wavenumber $k_{\mathrm{u}\mathrm{p}\mathrm{p}}^{\mathrm{a}}$ of the fluctuations (m⁻¹) is defined by the reciprocal of $a^{\mathrm{a}}$, i.e.,

$$k_{\mathrm{u}\mathrm{p}\mathrm{p}}^{\mathrm{a}}\equiv {\displaystyle \frac{2\pi }{a^{\mathrm{a}}}} \left(>0\right)$$

(90)

The possible area condition to avoid singularity, i.e., the basic condition of vortex rotation shown in Equations (82a) and (82b) is thus supplied by

$$k_{\mathrm{u}\mathrm{p}\mathrm{p}}^{\mathrm{a}}<{\left({\displaystyle \frac{5}{8}}\right)}^{1/2}{\epsilon }^{\mathrm{a}}$$

(91)

where from Equation (71b), ${\epsilon }^{\mathrm{a}}$ is expressed as

$${\epsilon }^{\mathrm{a}}\equiv {\left\{{\displaystyle \frac{4\rho }{{\sigma }^{*}{\nu }^{\mathrm{a}}}}{\left({\displaystyle \frac{\mathsf{\Omega }}{B_0}}\right)}^2\right\}}^{1/2}$$

(92)

${\nu }^{\mathrm{a}}$ is the kinematic viscosity in the 1st generation.

Then, for the convenience of experiment, we rewrite the angular velocity $\mathsf{\Omega }$ (s⁻¹) by the rotational frequency $f$ (s⁻¹, Hz).

$$\mathsf{\Omega }=2\mathsf{\pi }f$$

(93)

From Equations (90)–(93), the critical condition to avoid singularity is therefore expressed as

$${\left({\displaystyle \frac{B_0}{f}}\right)}^2=A^{\mathrm{a}}$$

(94a)

or

$$\left|B_0\right|={A^{\mathrm{a}}}^{1/2}\left|f\right|$$

(94b)

where

$$A^{\mathrm{a}}\equiv \left({\displaystyle \frac{5}{2}}\right){\displaystyle \frac{\rho {a^{\mathrm{a}}}^2}{{\sigma }^{*}{\nu }^{\mathrm{a}}}}$$

(94c)

For the 1st generation, using ${\nu }^{\mathrm{a}}~{10}^{-6}$ m² s⁻¹, $a^{\mathrm{a}}~{10}^{-4}$ m with ${\sigma }^{*}~10 \mathsf{\Omega }$⁻¹ m⁻¹ and $\rho ~{10}^3$ kg m⁻³, we can estimate

$${A^{\mathrm{a}}}^{1/2}~1.0 \mathrm{T} \mathrm{s}$$

(94d)

Equation (94b) is approximately expressed by a straight line with a slope of 1 through the origin. In the upper region of Equation (94b), due to satisfying the basic condition of vortex rotation $k^{\mathrm{a}}<{\left(5/8\right)}^{1/2}{\epsilon }^{\mathrm{a}}$ from Equation (82a), as shown in Equations (81a) and (81b), the vorticity coefficients ${\gamma }_{\mathrm{r}}^{*}$ and ${\gamma }_{\mathrm{f}}^{*}$ do not change the signs, keeping positive, so that ACW and CW rotations on the rigid and free surfaces are permitted for vortices in all other generations triggered by the vortex rotations in the 1st generation. The basic condition for the RMHD vortex rotation in Equation (82a) is more explicitly rewritten as

$$\left|B_0\right|>1.0\left|f\right|$$

(95)

On the contrary, in the lower region $\left|B_0\right|<1.0\left|f\right|$ corresponding to $k^{\mathrm{a}}>{\left(5/8\right)}^{1/2}{\epsilon }^{\mathrm{a}}$, since the system always contains the singular point $k^{\mathrm{a}}={\left(3/4\right)}^{1/2}{\epsilon }^{\mathrm{a}}$, any RMHD vortex rotation induced by the precession is forbidden, and a relatively stationary solution is left on the electrode in the rotational system. Instead, in the abandoned stationary solution, a macroscopic tornado-like rotation, i.e., VMHDF will be newly formed to give rise to another type of chiral deposition depending on the direction of magnetic field polarity (odd chirality) [28]. Owing to system rotation, such a MED process would have higher chiral efficiency than ordinary MED in a stationary system; for the angular momentum conservation, the population of either of the ACW and CW vortices in precession could be increased with the system rotation.

For the vortices in all the generations to follow the same critical condition as Equation (94d), the kinematic viscosities in the 2nd and 3rd generations must be changed. Since the parameters characterizing the fluctuations in each generation are the autocorrelation distance and the kinematic viscosity, for the symmetrical fluctuations with an autocorrelation distance $a~{10}^{-7} \mathrm{m}$ in the 2nd generation, a kinematic viscosity satisfying Equation (94d) should be $\nu {~10}^{-12}$ m² s⁻¹. For the fluctuations in the 3rd generation, having an autocorrelation distance $a~{10}^{-10}$ m, we obtain a kinematic viscosity $\nu ~{10}^{-18}$ m² s⁻¹. Such apparent viscosity changes are adjusted by the vacancy layer, which can subordinately change the viscosity by iso-entropic vacancies.

Assuming kinematic viscosities of ${10}^{-12}$ m² s⁻¹ and ${10}^{-18}$ m² s⁻¹ for the 2nd and 3rd generations, we have the same critical condition as Equation (94d). For the MED under VMHDFs in stationary systems [19], however, the critical conditions for nano- and ultra-micro-vortexes to rotate give rise to kinematic viscosities, ${10}^{-12}$ m² s⁻¹ and ${10}^{-30}$ m² s⁻¹, respectively. They are much smaller than the present values. These differences come from the difference in ease for vortex rotation. In the present case, vortexes are directly made by the system rotation, whereas in the previous case, they are indirectly driven in a cascade mode by the upper-generation vortexes. Obviously, the former vortexes can rotate much more easily than the latter.

2.13. Precessional Motions of RMHD Vortices by the System Rotation

Because some other singular points still exist in the area of the basic condition of vortex rotation in Equation (82a) or a more explicit condition in Equation (95), either of the rigid- and free-surface vortices is prohibited from rotating. From the analysis of Coriolis vorticity, as shown in Equations (87a)–(87c) and (88a)–(88c), it has been clarified that the sign of the product $B_0\mathsf{\Omega }$ determines the precessional direction of the vortices. As shown in Figure 3, to conserve local angular momentum and mass, a pair of neighboring vortices with downward and upward flows on the rigid and free surfaces rotate each other in opposite directions. The problem is therefore focused on which is prohibited to rotate, and which resultantly starts precession.

Under the basic condition of vortex rotation in Equation (82a), either of them is accelerated by the precession in the direction indicated by $B_0\mathsf{\Omega }$, and the other rotating in the opposite direction is decelerated by the balance between Lorentz and Coriolis forces. The singular points are therefore not contained in the vortex filter function of the accelerated one, but are contained in that of the decelerated one. Whether the filter function contains singular points or not determines which vortex receives precession.

A singular point occurs when the amplitude of a fluctuation diverges to infinity or becomes zero. This means that around a singular point, the fluctuations cannot exist, or the phases of the fluctuations do 180-degree turns, so that, as shown in Figure 7, the corresponding vortices stop rotation. However, due to the conservation of local angular momentum and mass of the pair vortex rotation, it subordinately keeps the same rotation with the help of the other vortex in precession. However, as discussed above, if both of them had singular points at the same time, only a pseudo-stationary solution area would be left there. In the following, considering the application to RMED, we treat the precessional process in the case of cathodic reaction.

2.13.1. Singular Points of the Rigid-Surface Vortex (RV) for the Range $k<k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$

As shown in Figure 8, the numerator functions of the rigid- and free-surface filter functions of the RV and FV, ${\mathrm{g}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{g}}_{\mathrm{f}}\left(k\right)$ in Equations (73c) and (75c) have singular points at $k_{\mathrm{r}\mathrm{s}}=k_{\mathrm{f}\mathrm{s}}={\left(3/4\right)}^{1/2}\epsilon$, which are, however, outside the basic condition of vortex rotation $0<k<k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$, where $k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}{\equiv \left(5/8\right)}^{1/2}\epsilon$ is the limiting singular point shown in Equation (82a). Therefore, under the basic condition, we can restrict the examination of singular points to the denominator functions ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$.

For the rigid-surface case of RV, the denominator function is given by Equation (73b) as follows.

$${\mathrm{h}}_{\mathrm{r}}\left(k\right)=64k^5\left\{3-4{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}+\zeta \left(1+3{\xi }^2\right)$$

(96a)

where

$$\zeta \equiv z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{*}{T}^{*}{R}^{*}\left(\propto B_0\mathsf{\Omega }\right)$$

(96b)

In the present case of cathodic reaction, due to $R^{*}>0,$ the sign of $\zeta$ is equal to the sign of $B_0\mathsf{\Omega }.$ The derivative with regard to $k$ is obtained as

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}k}}{\mathrm{h}}_{\mathrm{r}}\left(k\right)=64k^4\left\{15-28{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}$$

(97)

Therefore, ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ is an upward convex function of $k$, and the maximum value is

$${\mathrm{h}}_{\mathrm{r}}\left(k_{\mathrm{r},\mathrm{m}\mathrm{a}\mathrm{x}}\right)={\displaystyle \frac{12}{7}}{\left({\displaystyle \frac{15}{7}}\right)}^{5/2}{\epsilon }^5+\zeta \left(1+3{\xi }^2\right)$$

(98a)

where $k_{\mathrm{r},\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum wavenumber.

$$k_{\mathrm{r},\mathrm{m}\mathrm{a}\mathrm{x}}={\left({\displaystyle \frac{15}{28}}\right)}^{1/2}\epsilon$$

(98b)

In addition, ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ has special values at $k=0$ and $k_0$, i. e.,

$${\mathrm{h}}_{\mathrm{r}}\left(0\right)={\mathrm{h}}_{\mathrm{r}}\left(k_0\right)=\zeta \left(1+3{\xi }^2\right)$$

(99a)

where ${\mathrm{h}}_{\mathrm{r}}\left(0\right)$ is a vertical intercept, and $k_0$ is equal to the singular points of ${\mathrm{g}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{g}}_{\mathrm{f}}\left(k\right)$, defined as

$$k_0\equiv {\left({\displaystyle \frac{3}{4}}\right)}^{1/2}\epsilon \left({=k}_{\mathrm{r}\mathrm{s}}{=k}_{\mathrm{f}\mathrm{s}}>k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}\right)$$

(99b)

$k_0$ is also outside the basic condition of vortex rotation. In addition, ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ makes parallel translation with $\zeta$, i.e., $B_0\mathsf{\Omega }$ in the vertical direction so that we examine the precessional motions according to the sign of $B_0\mathsf{\Omega }$ in Equations (87a)–(87c).

For $B_0\mathsf{\Omega }>0$ (ACW precession): In the present case, $\zeta >0$ is fulfilled, so that, as shown in Figure 9a, the vertical intercept ${\mathrm{h}}_{\mathrm{r}}\left(0\right)$ in Equation (99a) is kept positive, and due to the extreme dependence of the $k^5$-term on $k$, beyond ${k=k}_0$, ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ steeply decreases to the negative area, so that the single singular point $k_{\mathrm{r}\mathrm{s}0}$ (m⁻¹) corresponding to

$${\mathrm{h}}_{\mathrm{r}}\left(k_{\mathrm{r}\mathrm{s}0}\right)=64{k_{\mathrm{r}\mathrm{s}0}}^5\left\{3-4{\left({\displaystyle \frac{k_{\mathrm{r}\mathrm{s}0}}{\epsilon }}\right)}^2\right\}+\zeta \left(1+3{\xi }^2\right)=0$$

(100a)

is approximately equalized to $k_0$, i.e.,

$$k_{\mathrm{r}\mathrm{s}0}\approx {\left({\displaystyle \frac{3}{4}}\right)}^{1/2}\epsilon \left(=k_0>k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}\right)$$

(100b)

This is equal to the singular point, $k_{\mathrm{r}\mathrm{s}}$ and $k_{\mathrm{f}\mathrm{s}}$ of ${\mathrm{g}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{g}}_{\mathrm{f}}\left(k\right)$ discussed above, and outside the basic condition of vortex rotation. Therefore, the rigid-surface vortices (RVs) do not contain any singular points under the basic condition. That is, the ACW rotation of the rigid-surface vortex is permitted for $B_0\mathsf{\Omega }>0$.

For $B_0\mathsf{\Omega }<0$ (CW precession): Since $\zeta$ becomes negative, $\zeta <0$, the intercept shifts downward to the negative side, i.e., ${\mathrm{h}}_{\mathrm{r}}\left(0\right)<0$. As far as the maximum value takes a positive value, ${\mathrm{h}}_{\mathrm{r}}\left(k_{\mathrm{r},\mathrm{m}\mathrm{a}\mathrm{x}}\right)>0$, as shown in Figure 9b, two singular points $k_{\mathrm{r}\mathrm{s}1}$ and $k_{\mathrm{r}\mathrm{s}2}$ ($k_{\mathrm{r}\mathrm{s}1}$ $<$ $k_{\mathrm{r}\mathrm{s}2}$) take place, i.e.,

$${\mathrm{h}}_{\mathrm{r}}\left(k_{\mathrm{r}\mathrm{s}1}\right)={\mathrm{h}}_{\mathrm{r}}\left(k_{\mathrm{r}\mathrm{s}2}\right)=0$$

(101a)

where $k_{\mathrm{r}\mathrm{s}1}$ exists inside the basic condition of vortex rotation, i.e.,

$${0<k}_{\mathrm{r}\mathrm{s}1}<k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$$

(101b)

and in view of the rapid decrease of the ${\mathrm{k}}^5$-term, we also have

$$k_{\mathrm{r}\mathrm{s}2}\approx k_0\left(>k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}\right)$$

(101c)

which is outside the basic condition. The basic condition of vortex rotation in Equation (82a) contains the smaller singular point $k_{\mathrm{r}\mathrm{s}1}$, so that RV rotation is always prohibited for $B_0\mathsf{\Omega }<0$ under the basic condition.

2.13.2. Singular Points of the Free-Surface Vortex (FV) for the Range $k<k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$

For the free surface case of FV, from Equation (75b), the equation to examine singular points is provided as

$${\mathrm{h}}_{\mathrm{f}}\left(k\right)\equiv 32k^6\left(2+{3\xi }^2\right)\left\{3-4{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}-\zeta \left\{3-8{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}\left\{5-8{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}$$

(102)

The special values of ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$ at $k=0$ and $k_0$ are represented as

$${\mathrm{h}}_{\mathrm{f}}\left(0\right)=-15\zeta$$

(103a)

where ${\mathrm{h}}_{\mathrm{f}}\left(0\right)$ is the vertical intercept, and ${\mathrm{h}}_{\mathrm{f}}\left(k_0\right)$ is given by

$${\mathrm{h}}_{\mathrm{f}}\left(k_0\right)=-3\zeta$$

(103b)

where $k_0$ is the wavenumber defined by Equation (99b).

Since ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$ extremely depends on $k^6$, the function form is mainly determined by the first term on the right-hand side of Equation (102). The derivative with respect to k is approximated by this term.

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}k}}{\mathrm{h}}_{\mathrm{f}}\left(k\right)\approx 64\left(2+{3\xi }^2\right)k^5\left\{9-16{\left({\displaystyle \frac{k}{\epsilon }}\right)}^2\right\}$$

(104a)

Therefore, ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$ is an upward convex function of $k$, and the maximum value is

$${\mathrm{h}}_{\mathrm{f}}\left(k_{\mathrm{f},\mathrm{m}\mathrm{a}\mathrm{x}}\right)\approx {\displaystyle \frac{3}{8}}{\left({\displaystyle \frac{3\epsilon }{2}}\right)}^6+{\displaystyle \frac{3}{4}}\zeta$$

(104b)

where $k_{\mathrm{f},\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum wavenumber.

$$k_{\mathrm{f},\mathrm{m}\mathrm{a}\mathrm{x}}\approx {\displaystyle \frac{3}{4}}\epsilon$$

(104c)

In the same way as ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$, ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$ also makes a parallel translation with $\zeta$, i.e., the value of $B_0\mathsf{\Omega }.$ However, as shown in Figure 9, comparing the special values of ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$ with those of ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$, we can see that the dependence on $\zeta$ is opposite.

For $B_0\mathsf{\Omega }>0$ (ACW precession): Because of $\zeta >0$, ${\mathrm{h}}_{\mathrm{f}}\left(0\right)$ and ${\mathrm{h}}_{\mathrm{f}}\left(k_0\right)$ in Equations (103a) and (103b) take negative values, so that ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$ takes a function form similar to ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ for $B_0\mathsf{\Omega }<0$. As represented in Equation (105b), since the maximum value becomes positive ${\mathrm{h}}_{\mathrm{f}}\left(k_{\mathrm{f},\mathrm{m}\mathrm{a}\mathrm{x}}\right)>0$, the equation ${\mathrm{h}}_{\mathrm{f}}\left(k\right)=0$ has two roots, i.e., singular points between $k=0$ and $k_0$. One is the smaller one $k_{\mathrm{f}\mathrm{s}1}>0$, and the other is the larger one, due to a rapid decrease by the $k^6$-term, approximately expressed by $k_{\mathrm{f}\mathrm{s}2}\approx k_0>k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$, which is therefore approximately obtained by replacing ${\left(k/\epsilon \right)}^2$ in the 2nd term on the right-hand side of Equation (102) with $3/4$, i.e.,

$${\mathrm{h}}_{\mathrm{f}}\left(k_{\mathrm{f}\mathrm{s}2}\right)\approx 32{k_{\mathrm{f}\mathrm{s}2}}^6\left(2+{3\xi }^2\right)\left\{3-4{\left({\displaystyle \frac{k_{\mathrm{f}\mathrm{s}2}}{\epsilon }}\right)}^2\right\}-3\zeta =0$$

(105)

Due to $\left|\xi \right|\ll 1$, comparing Equation (105) with Equation (96a), we can conclude that the singular points of RV and FV outside $k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$ take similar values. This means that since the basic condition of vortex rotation, ${k<k}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$ contains the smaller singular point $k_{\mathrm{f}\mathrm{s}1}$, on the contrary to the rigid-surface case, the FV rotation is always forbidden for $B_0\mathsf{\Omega }>0.$

For $B_0\mathsf{\Omega }<0$ (CW precession): Due to $\zeta <0,$ the intercept ${\mathrm{h}}_{\mathrm{f}}\left(0\right)$ in Equation (103a) is kept positive, and in view of the steep dependence of the $k^6$-term on $k$, in the same way as the case of RV, a single singular point $k_{\mathrm{f}\mathrm{s}0}$ corresponding to ${\mathrm{h}}_{\mathrm{f}}\left(k_{\mathrm{f}\mathrm{s}0}\right)=0$ is approximately equalized to $k_0$ as shown in Equation (100b). The condition to avoid singular points is thus consistent with the basic condition ${k<k}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$. This implies that for $B_0\mathsf{\Omega }<0$, the free-surface rotation of FV is always allowed under the basic condition.

2.13.3. Singular Points of the Filter Functions for the Range $k>k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$

As shown in Equations (73a) and (75a), since the vortex filter functions ${\mathrm{f}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{f}}_{\mathrm{f}}\left(k\right)$ are composed of the numerator functions ${\mathrm{g}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{g}}_{\mathrm{f}}\left(k\right)$ and the denominator functions ${\mathrm{h}}_{\mathrm{r}}\left(k\right)$ and ${\mathrm{h}}_{\mathrm{f}}\left(k\right)$, they infinitely diverge around the singular points. They are exhibited in Figure 10, where since the function values are plotted in logarithmic scale, the negative values are neglected. Under the basic condition of vortex rotation ${k<k}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$, more explicitly $\left|B_0\right|>1.0\left|f\right|,$ the singular points are thus expressed by positive divergences. Outside the condition ${k>k}_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$, i.e., $\left|B_0\right|<1.0\left|f\right|$, singular points always exist with infinite divergence, so that in this area, RMHD vortex rotations are always prohibited. The enlarged plots of the singular points are represented in normal scale. Compared with the area of the singular point of the rigid-surface vortex (RV), that of the free-surface vortex (FV) is quite narrow like a $\mathsf{\delta }$-function. In view of the fact that singular point arises from the balance of Lorentz and Coriolis forces, this means that for the balance on the free surface without any friction, pinpoint accuracy is required. On the other hand, on the rigid surface, such a balance is diffused by friction, so that a broader area of singular point appears.

2.13.4. Occurrence of Homochirality

To satisfy the conservation of local angular momentum and mass, the vortices forbidden to rotate would subordinately start rotation in the opposite direction to the neighboring vortices in precession. As a result, as shown in Figure 11, we can conclude that whether the product $B_0\mathsf{\Omega }$ is positive or negative, in a cathodic reaction like RMED, only a pair of ACW-RV and CW-FV emerge under the basic condition $k<k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$.

As has been discussed in Figure 4, due to no relative motion between solution and active points of nucleation, FV cannot make contributions to chiral electrodeposition, so that, as shown in Figure 11, chiral deposition is obtained not from FV, but from RV. As a result, as long as the condition $k<k_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{s}}$ or more explicitly $\left|B_0\right|>1.0\left|f\right|$ is satisfied in RMEDs, chiral nuclei with L-activity arising from the ACW rigid-surface vortices are obtained. This result strongly indicates the origin of homochirality by RMHD vortices in rotational magnetoelectrochemistry.

Such a breakdown of chiral symmetry of RMHD vortices would occur not only in electrodeposition, but also in usual electrochemical reductions. Furthermore, such a breakdown could be restored by considering anodic reactions, as shown in Equations (88a)–(88c), where the completely inverted chirality emerges. In cathodic reactions, L-active RVs arise, whereas D-active RVs emerge in anodic reactions.

3. Experiment

The experimental verification of the theoretical prediction was first performed for the RMED of Cu electrodeposition on a Cu disk, which was then immediately provided for the voltammetry of the oxidation of L- or D-alanine in a NaOH solution [28,29].

In the electrochemical cell of the RMED experiments, a conventional three-electrode system was employed: a polycrystalline Cu disc working electrode with a diameter of 3.2 mm, a Cu plate counter electrode, and a Cu wire reference electrode. To keep the solution from gravitational convection, as shown in Figure 12a, the working electrode was set downward, since the solution near the electrode becomes lighter than the bulk solution during deposition. It should be noted that such a situation corresponds to the upside-down configuration of Figure 5. The copper deposit films were fabricated in a 50 mM CuSO₄ aqueous solution containing 0.5 M H₂SO₄ at an overpotential of −0.45 V. In order to reduce the influence of vertical MHD flows, the working electrode was embedded in a tube wall (see Figure 12b), and a pulse electrodeposition was conducted with a pulse width of 5 s. The total electric charge was 0.4 C cm⁻², and the film thickness was approximately 150 nm.

The RMED was conducted with a hand-made experimental system as shown in Figure 12a. The electrochemical cell was placed at the center of a cylindrical bore in a cryocooled superconducting solenoidal magnet (Sumitomo Heavy Industries Ltd., Tokyo, Japan) in the High Field Laboratory for Superconducting Materials (HFLSM), Institute for Materials Research (IMR), Tohoku University. A magnetic field was perpendicularly applied to the working electrode, and the upward (+) and downward ($-$) $B_0$ in Figure 12a, reversely correspond to the downward and upward $B_0$ in Figure 12b. The electrode was set at the maximum point of the parabolic distribution of magnetic flux density. The field uniformity was confirmed to be within $\pm 1\%$ over the electrode surface. The cell was rotated by a geared motor system with a bevel gear and a non-magnetic stainless-steel shaft. Because the geared motor was sensitive to stray magnetic fields, it was placed at a distance of 2 m from the magnet center, where the magnetic field was less than 5 mT. The rotations with 0.5–6 Hz frequencies were applied to clockwise (CW) $\left(\mathsf{\Omega }<0\right)$ or anticlockwise (ACW) $\left(\mathsf{\Omega }>0\right)$ direction (viewed from the top of the electrode).

To estimate the surface chirality of RMED films, the just-prepared films were used as electrodes after the pretreatment of a potential sweep from $-$0.3 to 0.3 V in a 0.1 M NaOH aqueous solution [28,29], and the voltammogram of L- or D-alanine was measured on such a film electrode. The voltammetric measurements were conducted in a 20 mM L- or D-alanine + 0.1 M NaOH aqueous solution with a potential sweep rate of 10 mV s⁻¹ in the absence of a magnetic field.

The chirality in voltammograms was evaluated by an enantiomeric excess (ee) ratio defined as

ee = (i_p^L − i_p^D)/(i_p^L + i_p^D)

(106)

where i_p^L and i_p^D denote the peak currents of L- and D-alanine, respectively. The positive sign of the ee ratio represents L-activity, and the negative one represents D-activity. The ee ratios were obtained by one experiment at each condition; thus, the errors are not reported. However, the RMED experiments were conducted more than 200 times in various magnetic fields and rotational frequencies to explore the systematic chiral behavior of the RMED films [29]. As shown in Appendix E, the absolute value of ee, i.e., $\left|ee\right|$ is equal to the evolution ratio of chiral screw dislocation by microscopic RMHD vortices.

4. Results and Discussion

The experimental results are summarized by the diagram of the enantiomeric activity of the copper surface formed by RMED in Figure 13 [29], where the absolute values of the magnetic flux density (T) and rotational frequency (Hz) are taken as the vertical and horizontal lines, respectively. A broad straight line passing through the origin indicates the critical condition of RMHD vortex rotations, $\left|B_0\right|={A^{\mathrm{a}}}^{1/2}\left|f\right|$ in Equation (94b) for ${A^{\mathrm{a}}}^{1/2}=$1.0 T s. The areas of types II and III of enantiomeric activity around the line, therefore, imply the intermediate zones. Type IV and type I areas correspond to those clearly satisfying the conditions $\left|B_0\right|>1.0\left|f\right|$ and $\left|B_0\right|<1.0\left|f\right|$, respectively; in the type IV area allowing RMHD vortex rotation, as discussed above, the RMED surfaces show homochirality with L-activity, while in the type I area prohibiting RMHD vortex rotation, a vertical MHD flow (VMHDF) newly emerges, inducing MHD vortices instead so that the chirality of the deposit surface is consistent with that of the MED with odd chirality by VMHDF [30]. In type II and type III areas, intermediate dependence on the directions of rotation and magnetic field appears; type II deposits exhibit chiral signs depending on the rotational direction. Type III area deposits provide a special chirality indicating that both effects of VMHDF and system rotation are superimposed [29].

In type I area, though RMED is forbidden, the angular momentum by the system rotation is applied to the solution in a relatively stationary state, so that the population of righthanded and lefthanded vortex fluctuations deviates from the standard state of 50% and 50% of absolutely stationary solution in accordance with the given rotational direction. As a result, in keeping such a population, the arising tornado-like rotation VMHDF yields a MED film with higher efficiency than that in an absolutely stationary solution. Though the data were scattered, under the conditions of $\pm 2$ T and $\pm 4$ Hz satisfying $\left|B_0\right|<1.0\left|f\right|$, at most ee = $-$0.33 for D-activity was obtained, of which absolute value greatly exceeds the theoretical upper limit of ee, as discussed in Appendix E, $1/2^3=0.125$ of a conventional MED by a VMHDF in an absolutely stationary solution [18]. This is attributed to the change in the population of the initial vortices by the system rotation. The same situation also appears in the RMED, where the system rotation directly transfers the ultra-micro vortices for chiral screw dislocations to increase the ee up to over 0.4 for L-activity [28]. These results strongly suggest that the chiral activities of deposit surfaces are attributed to chiral vortex rotation.

Since the RMED homochirality is determined only by the vortices controlled by the MHD conditions, it is irrelevant to electrochemical conditions such as specific adsorption of chloride ions. On the other hand, the MED from a VMHDF controlled by electrochemical processes changes the chirality of D-activity with chloride additive, providing L-activity [18,21]. This is because the unstable growths of 2D nucleus and screw dislocation in the 1st and 3rd generations are stabilized by the chloride adsorption. In both generations, the partner vortices receiving precessions change from the free-surface ones to the rigid-surface ones [19].

On the bottom of the deep sea, if under the conditions of the type IV area, enantiomeric amino acids were electrochemically reduced on the surfaces of ferromagnetic substances such as magnetite under seawater rotations, their L-active products would be predominant. Especially because the homochirality of the RMED consists of physical quantities such as magnetic field and system rotation, strong resistivity would be expected against any environmental change.

5. Conclusions

The theory of the chirality of microscopic vortices in rotational magneto-electrochemistry has been experimentally validated by the rotational magneto-electrodeposition (RMED). The chirality of a rotational magnetohydrodynamic (RMHD) vortex is transferred to a screw dislocation not on the free surface covered with ionic vacancies, but on the rigid surface of exposed solid surface because the relative motion between the vortex and the active points of reaction is essential to the distinction of chirality. Beyond the critical condition of the RMHD vortex rotation, in any combination of magnetic field polarization and rotational direction, i.e., whether $B_0\mathsf{\Omega }$ is positive or negative, only vortices with ACW rotation emerge on the rigid surface of the electrode so that only L-active screw dislocations arise from the vortices. However, since such a chiral process depends on the direction of the electrolysis current, in the case of anodic oxidation, like anodic metal dissolution, CW vortices on the rigid surfaces would be predominant, so that D-active pits with right-handed chirality would be activated. This implies the restoration of symmetry from the breakdown of chiral symmetry in rotational magnetoelectrochemistry. The emergence of homochirality is possible not only for metallic ions in metal deposition and dissolution, but also for any enantiomeric reagents active for microscopic vortex motion. To explore the more general homochiral conditions for the origin of life, the application of rotational magnetoelectrochemistry to enantioselectivity would be of great importance.