Theory of Chiral Electrodeposition by Micro-Nano-Vortexes under a Vertical Magnetic Field-2: Chiral Three-Dimensional (3D) Nucleation by Nano-Vortexes

Abstract

The contributions of magnetohydrodynamic (MHD) vortexes to chiral electrodeposition in a vertical magnetic field were theoretically examined based on the three-generation model of the 2D nucleus, 3D nucleus, and screw dislocation; for the vortexes to rotate in the second and third-generation, the kinematic viscosity must be at least 10⁻¹⁸ and 10⁻³⁰ times lower than the ordinary value in the first generation, i.e., almost equal to zero. This implies that the ionic vacancy created on the electrode surface works as an atomic-scale lubricant. At the same time, the vortexes played three roles: promotion and suppression of nucleation, and transport of the chirality from the upper generation to the lower generation through precessional motion. Then, the rule of the chirality transfer was established, and finally, the relationship between the chiral activity and magnetic field was clarified in the presence and absence of chloride ions.

1. Introduction

In an electrode reaction under a vertical magnetic field, a tornado-like macroscopic rotation called vertical magnetohydrodynamic (MHD) flow (VMHDF) appears on a disk electrode (vertical MHD electrode (VMHDE)). On the electrode surface under a VMHDF, numerous microscopic vortexes called micro- and nano-MHD flows are formed, accompanied by the rotations of microbubble clusters originating from ionic vacancies produced by electrode reactions [1,2,3].

Ionic vacancies are created for conserving the linear momentum and electricity in the electron transfer of an electrode reaction, and initial naked vacancies, i.e., embryo vacancies, are formed in a free space [4,5]. Charged particles such as embryo vacancies are energetically unstable in a solvent, so they are quickly solvated by emitting solvation energy and enveloped by ionic clouds. Using the energy, ionic vacancies make their cores become enlarged, so the solvation energy is stored in them without entropy production. As shown in Figure 1, a vacancy core of about 0.1 nm is composed of a free space surrounded by polarized solvent molecules. An ionic vacancy migrates among solvent molecules by repeating shrinking and expanding reversibly without entropy production. This means that an ionic vacancy does not interact with surrounding solvent molecules, working as an atomic-scale lubricant. In other words, there is no energy dispersion in this diffusional motion. This is experimentally validated because the natural lifetime of an ionic vacancy is about 1 s [6,7], which is extraordinarily long compared with the collision frequency of solvent molecules of 10⁻¹⁰ s. Therefore, a vacancy layer formed under a VMHDF is expected to have superfluidity similar to that of liquid helium.

Copper electrodeposition under a vertical magnetic field gives rise to chiral screw dislocations with remarkable chiral activity caused by a VMHDF [1]. However, for the formation of chiral screw dislocations of the order of 0.1 nm from a VMHDF with a diameter of 1 cm, size control over eight figures was directly required, which seemed impossible. To solve such a difficulty, the three-generation model of chiral screw dislocation was proposed in Part 1 [1]; chiral two-dimensional (2D) nuclei arise from chiral vortexes of micro-MHD flows (~0.1 mm) as the first generation, then chiral three-dimensional (3D) nuclei are formed by chiral vortexes of nano-MHD flows (~0.1 μm) as the second generation, and finally, chiral screw dislocations are created by chiral ultra-micro vortexes (~0.1 nm) as the third generation. As shown in Figure 2, all three steps constitute a nesting box structure. The validity of this model was proven by the fact that a measure of the chiral activity, i.e., the enantiomeric excess (ee) ratio, cannot exceed a limiting value of 0.125 [1].

Chiral vortexes in each generation are, as shown in Figure 3a, produced by a two-layer system; the upper layer rotates around a vertical axis, driven by the bottom of a vortex on a rigid-surface nucleus of the upper generation, which, in the present second generation, corresponds to the bottom of the micro-MHD vortex on a rigid-surface of the 2D nucleus. In contrast, the lower layer is kept stationary by the pinning effect of the frictions of the rigid-surface vortexes, i.e., nano-MHD vortexes on the rigid-surface 3D nuclei. The lower-layer vortexes are reproduced with the same rotations as the upper-layer vortexes through the boundary between the two layers. Some upper-layer vortexes revolve with the upper layer, receiving precession from the upper-layer rotation. Such precessional motion transfers back to the vortexes in the lower layer. As a whole, they form a positive feedback cycle between the two layers.

Here, the vortexes in the lower layer are composed of a pair of vortexes with rigid and free surfaces, which, due to local linear momentum conservation, correspond to downward and upward flows, respectively. As shown in Figure 3b,c, an upward flow gathers ionic vacancies produced at the electrode, making a free surface covered with ionic vacancies without friction, whereas a downward flow blows off the ionic vacancies, exposing a bare surface with friction. At the same time, due to the continuity of vortexes, i.e., the local conservation of angular momentum, they have anticlockwise (ACW) and clockwise (CW) rotations, and vice versa. Figure 3d represents the case of upward (free-surface) ACW and downward (rigid-surface) CW vortexes.

Which vortexes, ACW or CW, receive the precession depends on whether they are unstable or not; if they are unstable, the vortex without friction on the free surface develops faster than the vortex with friction on the rigid surface, so the free-surface vortex (FV) will receive the precession. The rigid-surface vortex (RV) subordinately rotates in the opposite direction. On the contrary, when they are stable, though the vortex without friction on the free surface quickly disappears, due to friction, the vortex on the rigid surface dwindles more slowly. This time, the rigid surface vortex will gain the precessional motion. The chirality transfer of the vortexes between the two layers shown in Figure 3a represents the unstable case under an upward magnetic field, i.e., the ACW rotation of the upper-layer provided by a rigid-surface micro-MHD vortex transfers to a free-surface nano-MHD vortex in the lower layer, so a rigid-surface nano-MHD vortex obtains a CW rotation, forming a CW 3D nucleus of the second generation.

As discussed above, chiral nuclei of the lower generation are created by the rigid-surface vortexes of the same generation under a rigid-surface vortex of the upper generation. Namely, chirality transfers from the upper generation to the lower generation through rigid-surface vortexes. Under the free-surface vortexes, the nucleation proceeds with the revolution of the vacancy layer on the electrode surface, so that no chiral nuclei emerge due to the lack of relative rotation. In the unstable case, the chiral rotations of the rigid-surface vortexes of the upper generation are transferred to the free-surface vortexes of the lower generation. However, the chirality of the lower-generation nucleus is determined by the rigid-surface vortex of the same generation, so the chirality is changed. On the contrary, in the stable case, the chirality of the rigid-surface vortexes of the upper generation directly transfers to the rigid-surface ones of the lower generation, i.e., the chirality is preserved.

In chiral deposition, vortexes in various scales play important roles. Here, we come up against a serious problem; the smaller the vortex is, the more difficult it is to rotate. In short, much smaller vortexes cannot exist due to increasing effective viscosity. However, fortunately, since ionic vacancies work as atomic-scale lubricants and form a vacancy layer with a thickness of around 1 μm, 3D nuclei with a size of about 0.1 μm are prepared by rigid-surface nano-MHD vortexes in a liquid with extremely low viscosity. As will be discussed later, theoretical calculation allows us to expect that the vortexes over 3D nuclei can rotate in a liquid with a kinematic viscosity less than 10⁻¹⁸ m² s⁻¹, which is 10⁻¹² times smaller than the ordinary one (~10⁻⁶ m² s⁻¹). This means that the ultra-micro vortexes for chiral screw dislocations require almost zero viscosity.

Chiral magnetoelectrodeposition was first found by Mogi [8]; he and his coworkers have been experimentally clarifying the various aspects of the chirality evolution under magnetic fields [9,10,11,12,13,14,15,16,17,18,19,20,21], e.g., the dependence on magnetic field intensity, the effect on the presence and absence of chloride ions, and even the application in rotating systems. Based on these experimental results, in this paper, we theoretically examine the mechanism through which chiral activity arises from the various microscopic vortexes under a vertical magnetic field. As for magneto-electrochemistry, for about five decades, many researchers have been making great efforts to establish its foundations; electrode reactions under magnetic fields inevitably induce Lorentz forces, leading to many characteristic applications. For example, in calorimetry experiments, it is critically important to keep an experimental system iso-thermal. In this point, magneto-electrochemistry has a great advantage because the Lorentz force induces an MHD flow, in which mixing motion easily accomplishes a uniform distribution of temperature. Recently, using an electrode with a rectangular channel for an MHD flow called an MHD electrode, we succeeded in measuring the excess heat arising from the pair annihilation of ionic vacancies with opposite charges created at the cathode and anode [22,23,24]. The measured heat amounts were in good agreement with the theoretical calculation. In those cases, the Lorentz force not only mixes the ionic vacancies but also guides them to collision.

Several useful tools have been also developed for reaction analyses in magnetic fields [25,26,27,28,29,30,31,32]. Fahidy commented that an MHD flow decreases the thickness of a diffusion layer, enhancing mass transfer in an electrode reaction [29,30,31]. Olivier theoretically examined the MHD effect on microelectrodes [32,33,34,35] and conducted electrochemical impedance spectroscopy in a magnetic field [36,37]. White studied the MHD flow of ultra-micro-disk-electrodes in non-aqueous systems containing organic reactants [38,39,40]. Using flow visualization techniques in a magnetic field, Mutschke and co-workers examined electrodeposition in cuboid cells under magnetic fields accompanied by 3D convections affected by a gravitational field, which were compared with numerical simulations [41,42]. The contributions of the MHD flow to the phase compositions of composite metals have been investigated by many researchers (Oliver, Alemany, Daltin, Chopart, Hinds, Coey, and Zabiński) [43,44,45,46,47,48,49,50,51]. As a remarkable magneto-electrochemical effect, we can cite the magneto-convection by the gradient field force under a heterogeneous magnetic field, which promotes the mass transfer process in an electrode reaction [52,53,54]. Tschulik, Uhlemann, and Mutshchke, as well as Dunne and Coey, clarified that a superimposed Lorentz force donates more complicated effects to deposit patterns and compositions [55,56,57,58,59]. As mentioned above, effective MHD flows under parallel magnetic fields are provided by the MHD electrode (MHDE) [60,61,62]. The velocity and concentration distributions of the electrode are expressed by simple equations of velocity and the limiting diffusion current. As Fahidy said, the agreement between the theory and the experimental result is excellent [29].

In recent days, various studies on the transport of microscopic materials in solutions by using MHD forces have been reported. Kuhn et al. proposed an efficient alternative mechanism with which to power self-electrophoretic Mg/Pt Janus swimmers based on the Lorentz force and designed a self-propelled bimetallic Janus rotor and a wireless magnetoelectrochemical rotor [63,64,65]. Celzard et al. proposed self-propelled particles based on the MHD acceleration of the surrounding fluid [66]. Fritsch et al. designed analysis systems with microfluidics based on the redox MHD concept and proposed a method for combining dark-field microscopy with self-contained redox MHD microfluidics to facilitate single-particle analysis in mixtures of dispersion single-nanoparticle populations [67,68].

In the present paper, we first examine the similarity of the various MHD vortexes in the three generations, and derive the relationship between the representative length and the kinematic viscosity, determining the effective values of viscosity in each generation. Then, we establish the instability equations of 3D nucleation under nano-MHD vortexes in the second generation. Finally, using the equations, we present a general method to determine the chirality under a VMHDF in the absence and presence of chloride ions.

2. Theory

Briefly, 3D nuclei grow on a 2D nucleus formed in an electric double layer, developing the diffusion layer outside the electric double layer [69,70,71]. As the reaction proceeds, the physical quantities concerned change, approximately expressed by symmetrical fluctuations, i.e., changes around their average values [71].

For example, the concentration fluctuation of the metallic ion is defined by the following:

$$c_{\mathrm{m}}{\left(x,y,z,t\right)}^{\mathrm{s}}\equiv C_{\mathrm{m}}\left(x,y,z,t\right)-〈C_{\mathrm{m}}\left(x,y,z,t\right)〉$$

(1)

where superscript “s” implies a symmetrical fluctuation. $C_{\mathrm{m}}\left(x,y,z,t\right)$ and $〈C_{\mathrm{m}}\left(x,y,z,t\right)〉$ are the molar concentration and its average value, respectively. The sign “$〈 〉$” means the average over the electrode surface. As shown in Figure 4a, at the equilibrium potential, $E=E_{\mathrm{e}\mathrm{q}}$, the concentration itself and the concentration gradient fluctuate toward the positive and negative sides of their equilibrium values. However, when the electrode potential deviates to a cathodic direction, $E<E_{\mathrm{e}\mathrm{q}}$, as shown in Figure 4b, the electrodeposition proceeds on 3D nuclei and they one-sidedly fluctuate. Only negative and positive components of the fluctuations survive. This means that in a non-equilibrium state, either side of the amplitude of the equilibrium fluctuation is cut off, which depends on whether the cathodic or anodic phase cutting of symmetrical fluctuations occurs. The phase cutting is, as will be shown later in computer graphics, important for creating a cut surface on the 3D nucleus without roughening via diffusion.

As for vortexes, as exhibited in Figure 3, vortexes having downward and upward flows correspond to rigid- and free-surface vortexes, respectively. That is, the signs of the z-components of the velocities of the rigid- and free-surface vortexes are fixed as $w_{\mathrm{r}}^{\mathrm{s}}<0$ and $w_{\mathrm{f}}^{\mathrm{s}}>0$, respectively, where the subscripts “r” and “f” indicate rigid- and free-surfaces. Figure 3d indicates that to conserve the local angular momentum, a pair of vortexes with upward and downward flows rotate in opposite directions to each other, which means that the z-components of the vorticities of a vortex pair must have different signs, ${\omega }_{\mathrm{z},\mathrm{r}}^{\mathrm{s}}<0$ and ${\omega }_{\mathrm{z},\mathrm{f}}^{\mathrm{s}}>0$ or ${\omega }_{\mathrm{z},\mathrm{r}}^{\mathrm{s}}>0$ and ${\omega }_{\mathrm{z},\mathrm{f}}^{\mathrm{s}}<0$. Adopting a right-handed system with an upward magnetic field, we can define the negative and positive z-components of the vorticities as CW and ACW rotations. As a result, as shown in Figure 5, the vortexes in the lower layer are simply classified into two sets, where the phase-cutting quantities in the x-direction are schematically exhibited.

2.1. Similarity of the MHD Vortexes in the Three Generations

2.1.1. Vorticity Coefficient Functions (VCF)

The motion of MHD vortexes under a VMHDF is described by the vorticity coefficient function (VCF) established in Part 1 [1], which works as the characteristic function of the lower generation vortexes, controlling vortex instability. Using the functions, we can examine the similarity of each generation of vortexes. From now on, for similarity, we use the nondimensional wavenumber as “a” without superscript a and s, implying asymmetrical and symmetrical fluctuations in the first and second generations.

For rigid-surface vortexes, we have the three-generation functions:

$${\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)\equiv \frac{16Q^{\ast \mathrm{i}}{\mathrm{g}}_4\left(a\right)}{16Q^{\ast \mathrm{i}}{\mathrm{g}}_5\left(a\right)+R^{\ast \mathrm{i}}{S}^{\ast \mathrm{i}}{T}^{\ast \mathrm{i}-1}{\mathrm{g}}_6\left(a\right)} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{i}=1\mathrm{s}\mathrm{t},2\mathrm{n}\mathrm{d} \mathrm{a}\mathrm{n}\mathrm{d} 3\mathrm{r}\mathrm{d}$$

(2)

The function is expressed by the ratio of the torque component from the Lorentz force to the total torque of the vortexes. Here, the coefficients $S^{\ast \mathrm{i}}$, $T^{\ast \mathrm{i}}$, $Q^{\ast \mathrm{i}}$ and $R^{\ast \mathrm{i}}$ are defined by the following:

$$S^{\ast \mathrm{i}}\equiv \frac{B_0{d}^{\mathrm{i}}}{\rho {\nu }^{\mathrm{i}}}$$

(3a)

$$T^{\ast \mathrm{i}}\equiv \frac{2{\tilde{\mathsf{\Omega }}}^{\mathrm{j}}{d}^{\mathrm{i}}}{{\nu }^{\mathrm{i}}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{n}\mathrm{o}\mathrm{n},\mathrm{a} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{s}$$

(3b)

$$Q^{\ast \mathrm{i}}\equiv \frac{{\sigma }^{\mathrm{*}}{B_0}^2{d^{\mathrm{i}}}^2}{\rho {\nu }^{\mathrm{i}}} (>0)$$

(3c)

and

$$R^{\ast \mathrm{i}}\equiv \frac{L_{\mathrm{m}}{d^{\mathrm{i}}}^2}{D_{\mathrm{m}}} (>0)$$

(3d)

where for $\mathrm{i}=$ 1st, ${\nu }^{\mathrm{i}}$ denotes the ordinary kinematic viscosity of the electrolyte solution (${\nu }^{\mathrm{i}}\approx {10}^{-6} {\mathrm{m}}^2 {\mathrm{s}}^{-1})$, and for $\mathrm{i}=$ 2nd and 3rd, ${\nu }^{\mathrm{i}}$ denotes the effective kinematic viscosity of a vacancy layer formed on the electrode. As has been discussed in the introduction, due to atomic-scale lubricants, for ionic vacancies, the value greatly decreases down to zero. ${\tilde{\mathsf{\Omega }}}^{\mathrm{j}}$ is the representative angular velocity of the rigid-surface vortexes of the upper generation, i.e., $\mathrm{j}=$ non, a and s indicate VMHDF, micro-MHD vortexes, and nano-MHD vortexes, respectively. $B_0$ is the external magnetic flux density, $L_{\mathrm{m}}$ is the concentration gradient, and $D_{\mathrm{m}}$ is the diffusion coefficient of metallic ions. ${\sigma }^{\mathrm{*}}$ is the electric conductivity, $\rho$ is the solution density, and $d^{\mathrm{i}}$ is the representative length of the i-generation.

Then, from Part 1 [1], ${\mathrm{g}}_4\left(a\right)$, ${\mathrm{g}}_5\left(a\right)$ and ${\mathrm{g}}_6\left(a\right)$ are given by the following:

$${\mathrm{g}}_4\left(a\right)\equiv a^4\left({\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2 a+a^2\right)$$

(4a)

$${\mathrm{g}}_5\left(a\right)\equiv a^3\left({\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^2 a+a^2\right)$$

(4b)

$${\mathrm{g}}_6\left(a\right)\equiv z_{\mathrm{m}}F{D}_{\mathrm{m}}{\mathrm{e}}^a\left[2Q^{\ast \mathrm{i}}{a}^2\left\{5\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+2a \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a+\left(1+a\right){\mathrm{e}}^a\right\}+{d^{\mathrm{i}}}^2{T^{\mathrm{*}\mathrm{i}}}^2\left(1+a\right)\left(3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a+2a \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+a {\mathrm{e}}^a\right)\right]$$

(4c)

where $z_{\mathrm{m}}$ is the charge number of the deposit ion, and F is the Faraday constant.

For free-surface vortexes, we obtain the following:

$${\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)\equiv \frac{16Q^{\ast \mathrm{i}}{\mathrm{g}}_1\left(a\right)}{16Q^{\ast \mathrm{i}}{\mathrm{g}}_2\left(a\right)+R^{\ast \mathrm{i}}{S}^{\ast \mathrm{i}}{{T^{\ast \mathrm{i}}}^{-1}\mathrm{g}}_3\left(a\right)}$$

(5)

where ${\mathrm{g}}_1\left(a\right)$, ${\mathrm{g}}_2\left(a\right)$ and ${\mathrm{g}}_3\left(a\right)$ are given by the following:

$${\mathrm{g}}_1\left(a\right)\equiv a^3\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+a\right)$$

(6a)

$${\mathrm{g}}_2\left(a\right)\equiv a^2\left(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+a\right)$$

(6b)

$${\mathrm{g}}_3\left(a\right)\equiv z_{\mathrm{m}}F{D}_{\mathrm{m}}{\mathrm{e}}^a\left\{2Q^{\ast \mathrm{i}}a \left(5\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a+2a \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a\right)+{d^{\mathrm{i}}}^2{T^{\mathrm{*}\mathrm{i}}}^2\left(3\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+2a \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a\right)\right\}$$

(6c)

The vortex rotations are controlled by the second terms of the denominators of Equations (2) and (5) arising from the precession by an upper-generation rigid-surface vortex, and the similarity of the vortexes of the three generations depends on the representative length, $d^{\mathrm{i}}$, and the kinematic viscosity, ${\nu }^{\mathrm{i}}$.

In the following, to examine chiral similarity, we extract $d^{\mathrm{i}}$ and ${\nu }^{\mathrm{i}}$ from Equations (2) and (5). For simplicity, except for the sign, the angular velocity of the upper-generation rigid-surface vortex, ${\tilde{\mathsf{\Omega }}}^{\mathrm{j}}$, is equalized to that of a VMHDF, ${\tilde{\mathsf{\Omega }}}^{\mathrm{j}}=\tilde{\mathsf{\Omega }}$. Then, the following parameters are rewritten as follows:

$$S^{\ast \mathrm{i}}=S^{\ast }{\mathsf{\xi }}^{\mathrm{i}}$$

(7a)

$$T^{\ast \mathrm{i}}=T^{\ast }{\mathsf{\xi }}^{\mathrm{i}}$$

(7b)

$$Q^{\ast \mathrm{i}}=Q^{\ast }{d}^{\mathrm{i}}{\mathsf{\xi }}^{\mathrm{i}}$$

(7c)

$$R^{\ast \mathrm{i}}=R^{\ast }{d^{\mathrm{i}}}^2$$

(7d)

where ${\mathsf{\xi }}^{\mathrm{i}}$ is defined by

$${\mathsf{\xi }}^{\mathrm{i}} \equiv \frac{d^{\mathrm{i}}}{{\nu }^{\mathrm{i}}}$$

(7e)

and the parameters $S^{\ast }, T^{\ast }$, $Q^{\ast }$ and $R^{\ast }$ are defined by

$$S^{\ast }\equiv \frac{B_0}{\rho }$$

(8a)

$$T^{\ast }\equiv 2\tilde{\mathsf{\Omega }}$$

(8b)

$$Q^{\ast }\equiv \frac{{\sigma }^{\mathrm{*}}{B_0}^2}{\rho }$$

(8c)

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

(8d)

Therefore, we obtain the relationship:

$$R^{\ast \mathrm{i}}{S}^{\ast \mathrm{i}}{T^{\ast \mathrm{i}}}^{-1}=R^{\ast }{S}^{\ast }{T^{\ast }}^{-1}{d^{\mathrm{i}}}^2$$

(9)

For rigid-surface vortexes, the function controlling vortex rotations ${\mathrm{g}}_6\left(a\right)$ is expressed by the following:

$${\mathrm{g}}_6\left(a\right)=z_{\mathrm{m}}F{D}_{\mathrm{m}}{\mathrm{e}}^a\left[2Q^{\ast }{\mathrm{h}}_{\mathrm{m}\mathrm{r}}\left(a\right)d^{\mathrm{i}}{\mathsf{\xi }}^{\mathrm{i}}+{T^{\mathrm{*}}}^2{\mathrm{h}}_{\mathrm{p}\mathrm{r}}\left(a\right){\left(d^{\mathrm{i}}{\mathsf{\xi }}^{\mathrm{i}}\right)}^2\right]$$

(10)

The function ${\mathrm{h}}_{\mathrm{m}\mathrm{r}}\left(a\right)$ corresponds to the torque component of the rigid-surface MHD vortexes activated by the applied magnetic field, whereas ${\mathrm{h}}_{\mathrm{p}\mathrm{r}}\left(a\right)$ is the torque component of the precession from the upper-generation rigid-surface MHD vortexes.

$${\mathrm{h}}_{\mathrm{m}\mathrm{r}}\left(a\right)\equiv a^2\left\{5\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+2a \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a+\left(1+a\right){\mathrm{e}}^a\right\}$$

(11a)

and

$${\mathrm{h}}_{\mathrm{p}\mathrm{r}}\left(a\right)\equiv \left(1+a\right)\left(3\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a+2a \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+a {\mathrm{e}}^a\right)$$

(11b)

As a result, the function in Equation (2) is rewritten as follows:

$${\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)=\frac{16Q^{\ast }{\mathrm{g}}_4\left(a\right)}{16Q^{\ast }{\mathrm{g}}_5\left(a\right)+R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_6{\left(a\right)}^{\mathrm{i}}}$$

(12)

Namely, the similarity of vortex rotations between the three generations is concentrated to ${\tilde{\mathrm{g}}}_6{\left(a\right)}^{\mathrm{i}},$ which determines the lower-generation vortex motion, expressed by the following:

$${\tilde{\mathrm{g}}}_6{\left(a\right)}^{\mathrm{i}}\equiv z_{\mathrm{m}}F{D}_{\mathrm{m}}{\mathrm{e}}^a\left[2Q^{\ast }{\mathrm{h}}_{\mathrm{m}\mathrm{r}}\left(a\right){d^{\mathrm{i}}}^2+{T^{\mathrm{*}}}^2{\mathrm{h}}_{\mathrm{p}\mathrm{r}}\left(a\right)\left({d^{\mathrm{i}}}^4/{\nu }^{\mathrm{i}}\right)\right]$$

(13)

As $d^{\mathrm{i}}$ decreases with lower generations, Equation (13) reveals that the contribution from the magnetic activation of the vortexes also decreases. To maintain a similar vortex motion though decreasing $d^{\mathrm{i}}$, ${\tilde{\mathrm{g}}}_6{\left(a\right)}^{\mathrm{i}}$ in Equation (12) must therefore be kept in the same function form, i.e., the value of ${d^{\mathrm{i}}}^4/{\nu }^{\mathrm{i}}$ must be almost the same in any generation.

For free-surface vortexes, the function of the vortex rotation, ${\mathrm{g}}_3\left(a\right)$, in Equation (6c) is expressed as follows:

$${\mathrm{g}}_3\left(a\right)=z_{\mathrm{m}}F{D}_{\mathrm{m}}{\mathrm{e}}^a\left[2Q^{\ast }{\mathrm{h}}_{\mathrm{m}\mathrm{f}}\left(a\right)d^{\mathrm{i}}{\mathsf{\xi }}^{\mathrm{i}}+{T^{\mathrm{*}}}^2{\mathrm{h}}_{\mathrm{p}\mathrm{f}}\left(a\right){\left(d^{\mathrm{i}}{\mathsf{\xi }}^{\mathrm{i}}\right)}^2\right]$$

(14)

The function ${\mathrm{h}}_{\mathrm{m}\mathrm{f}}\left(a\right)$ is the magnetic field component of the free-surface vortex torque, and ${\mathrm{h}}_{\mathrm{p}\mathrm{f}}\left(a\right)$ is the precession component. Namely, we have the following:

$${\mathrm{h}}_{\mathrm{m}\mathrm{f}}\left(a\right)\equiv a \left(5\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a+2a \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a\right)$$

(15a)

and

$${\mathrm{h}}_{\mathrm{p}\mathrm{f}}\left(a\right)\equiv 3\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} a+2a \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} a$$

(15b)

As a result, the function in Equation (5) is rewritten as follows:

$${\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)=\frac{16Q^{\ast }{\mathrm{g}}_1\left(a\right)}{16Q^{\ast }{\mathrm{g}}_2\left(a\right)+R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_3{\left(a\right)}^{\mathrm{i}}}$$

(16)

In the same way as Equation (13), the similarity is concentrated to ${\tilde{\mathrm{g}}}_3{\left(a\right)}^{\mathrm{i}}$, i.e.,

$${\tilde{\mathrm{g}}}_3{\left(a\right)}^{\mathrm{i}}\equiv z_{\mathrm{m}}F{D}_{\mathrm{m}}{\mathrm{e}}^a\left[2Q^{\ast }{\mathrm{h}}_{\mathrm{m}\mathrm{f}}\left(a\right){d^{\mathrm{i}}}^2+{T^{\mathrm{*}}}^2{\mathrm{h}}_{\mathrm{p}\mathrm{f}}\left(a\right)\left({d^{\mathrm{i}}}^4/{\nu }^{\mathrm{i}}\right)\right]$$

(17)

For free-surface vortexes to keep the similarity, the term ${d^{\mathrm{i}}}^4/{\nu }^{\mathrm{i}}$ must be almost the same in all generations. The characteristic functions, VCFs of the MHD vortexes in Equations (12) and (16), indicate the ratio of the Lorentz force–torque component to the total torque. The total torques of the denominators in Equations (12) and (16) are composed of Lorentz force torque and induced vortex torques. Concerning the vortex torques, as shown in Equations (13) and (17), the MHD vortexes are induced via magnetic activation and the precession by the rotation of the upper-generation rigid-surface MHD vortex.

The induced vortex-torque component on the rigid surface with friction is larger than that on the free surface without friction. Namely, we have the following relationship:

$$\left[R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_3{\left(a\right)}^{\mathrm{i}}\right]<\left[R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_6{\left(a\right)}^{\mathrm{i}}\right]$$

(18)

This relationship leads to the difference between the rigid- and free-surface vortex motion.

2.1.2. Occurrence of Chiral Symmetry

The induced vortex torques change the sign depending on the rotational direction of the upper-generation rigid-surface vortex, $\tilde{\mathsf{\Omega }}$, which is more generally expressed by $B_0\tilde{\mathsf{\Omega }}>0$ for a right-handed system and $B_0\tilde{\mathsf{\Omega }}<0$ for a left-handed system. Namely, because the term $S^{\ast }{T}^{\ast -1}$ is proportional to $B_0{\tilde{\mathsf{\Omega }}}^{-1}$, the functions in Equations (12) and (16) are continuous for wavenumber $a$ for the right-handed system, $B_0\tilde{\mathsf{\Omega }}>0$, whereas they have singular points of $a$ for the left-handed system, $B_0\tilde{\mathsf{\Omega }}<0.$

However, considering the case where the induced vortex torques are dominant over the Lorentz force torque, we can make the singular point disappear. Equations (12) and (16) are therefore expressed as follows:

$${\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)\approx \frac{16Q^{\ast }{\mathrm{g}}_4\left(a\right)}{R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_6{\left(a\right)}^i}$$

(19)

and

$${\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)\approx \frac{16Q^{\ast }{\mathrm{g}}_1\left(a\right)}{R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_3{\left(a\right)}^i}$$

(20)

Therefore, for the right-handed system, we obtain

$${\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)>0 \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)>0$$

(21a)

and for the left-handed system, we obtain

$${\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)<0 \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)<0$$

(21b)

From Equation (18), the following relationship is satisfied:

$$\left|{\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)\right|<\left|{\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)\right|$$

(21c)

2.1.3. Conservation of Chiral Vortex Motion in the Three Generations

The representative length, $d^{\mathrm{i}}$, decreases with each generation. Under a constant kinematic viscosity, ${\nu }^{\mathrm{i}}$, in Equations (13) and (17), the induced vortex torque components $R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_6{\left(a\right)}^{\mathrm{i}}$ and $R^{\ast }{S}^{\ast }{T}^{\ast -1}{\tilde{\mathrm{g}}}_3{\left(a\right)}^{\mathrm{i}}$ converge to zero with decreasing $d^{\mathrm{i}}$. This is because the effective viscosity increases with decreasing representative length. Actually, in Equations (12) and (16), under a constant ${\nu }^{\mathrm{i}}$, as $d^{\mathrm{i}}$ approaches zero, we have

$${\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)\to \frac{16Q^{\ast }{\mathrm{g}}_4\left(a\right)}{16Q^{\ast }{\mathrm{g}}_5\left(a\right)}=a$$

(22a)

and

$${\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)\to \frac{16Q^{\ast }{\mathrm{g}}_1\left(a\right)}{16Q^{\ast }{\mathrm{g}}_2\left(a\right)}=a$$

(22b)

These results indicate that at the lower limit of $d^{\mathrm{i}}$, as will be shown later, the induced vortexes vanish.

In Equations (13) and (17), the induced vortex torque components are further separated by the magnetic activation and precession components, which decrease with ${d^{\mathrm{i}}}^2$ and ${d^{\mathrm{i}}}^4/{\nu }^{\mathrm{i}}$, respectively. For the induced vortexes to survive though decreasing $d^{\mathrm{i}}$, the kinematic viscosity, ${\nu }^{\mathrm{i}}$, must also decrease in proportion to ${d^{\mathrm{i}}}^4.$

In the case of the first-generation $\left(\mathrm{i}=1\mathrm{s}\mathrm{t}\right)$, for $d^{\mathrm{i}}={10}^{-4}$ m, an ordinary value for kinematic viscosity, ${\nu }^{\mathrm{i}}\approx {10}^{-6}$ m² s⁻¹ is assumed, so we have the following:

$$\frac{{d^{\mathrm{i}}}^4}{{\nu }^{\mathrm{i}}}\approx {10}^{-10}{\mathrm{m}}^2\mathrm{s}$$

(23)

According to this situation, in the second-generation $\left(\mathrm{i}=2\mathrm{n}\mathrm{d}\right)$ of $d^{\mathrm{i}}={10}^{-7}$ m, assuming ${\nu }^{\mathrm{i}}\approx {10}^{-18}$ m² s⁻¹, we obtain ${d^{\mathrm{i}}}^4/{\nu }^{\mathrm{i}}\approx$ ${10}^{-10}$ m² s. In the third generation $\left(\mathrm{i}=3\mathrm{r}\mathrm{d}\right)$ of $d^{\mathrm{i}}={10}^{-10}$ m, with ${\nu }^{\mathrm{i}}\approx {10}^{-30}$ m² s⁻¹, we derive the same value of ${d^{\mathrm{i}}}^4/{\nu }^{\mathrm{i}}\approx$ ${10}^{-10}$ m² s. As shown here, the actual high chiral activity obtained via copper deposition strongly requires a non-viscous vacancy layer to be formed on the electrode. In Figure 6, the VCFs in the three generations are plotted against the nondimensional wavenumber, $a$. As discussed in Equations (21a) and (21b), for the right-handed system, the VCFs take positive values, whereas the VCFs of the left-handed system become negative. Then, as pointed out in Equation (21c), the absolute value of the VCF on the free surface is larger than that of the VCF on the rigid surface. Under the present condition of Equation (23), the VCFs of the second and third generations are consistent with each other.

2.2. Characteristic Equations of 3D Nucleation under Nano-MHD Vortexes

In the second generation, the characteristic equations of 3D nucleation under nano-MHD vortexes control the instability of the 3D nucleation and nano-MHD vortexes, which are, as will be mentioned later, described by the characteristic equation of 3D nucleation in a stationary solution multiplied by the VCFs of the rigid-surface and free-surface nano-MHD vortexes. Therefore, we first examine the 3D nucleation in a stationary solution.

2.2.1. Surface Height Equation in 3D Nucleation

In Figure 7, the 3D nucleation process on the surface of a 2D nucleus in a stationary solution is exhibited. Assuming that the mass transfer is rate-determining, we will treat the electron transfer process in a quasi-equilibrium state. The mass balance between the adatoms and the metallic ions comes from the mass flux density, ${\overrightarrow{j}}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}^{\mathrm{s}}$, of the metallic ion from the solution phase to the electrode surface, the mass flux density of the adatom, ${\overrightarrow{j}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{\mathrm{s}}$, by the surface diffusion and the mass flux density, ${\overrightarrow{j}}_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\mathrm{s}}$, of the adatom by the incorporation into the crystal lattice [72,73,74,75].

$$\frac{\partial }{\partial t}{c}_{\mathrm{a}\mathrm{d}}{\left(x,y,t\right)}^{\mathrm{s}}+{\nabla }_{\perp }·{\overrightarrow{j}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{\mathrm{s}}=\overrightarrow{n}·{\overrightarrow{j}}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}^{\mathrm{s}}-\overrightarrow{n}·{\overrightarrow{j}}_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\mathrm{s}}$$

(24)

where $c_{\mathrm{a}\mathrm{d}}{\left(x,y,t\right)}^{\mathrm{s}}$ denotes the symmetrical fluctuation of the adatom concentration, $\overrightarrow{n}$ is the unit normal vector of the electrode surface, and ${\nabla }_{\perp }$ is defined by $\left(\partial /\partial x,\partial /\partial y\right)$.

Using the surface diffusion coefficient, $D_{\mathrm{a}\mathrm{d}}$, and the equilibrium surface concentration of the adatom, $C_{\mathrm{a}\mathrm{d}}^{\ast }$, we can write the mass flux density of the adatom along the surface as follows:

$${\overrightarrow{j}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{\mathrm{s}}=-\frac{D_{\mathrm{a}\mathrm{d}}}{RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{\nabla }_{\perp }\psi {\left(x,y,t\right)}^{\mathrm{s}}$$

(25)

where R and T are the universal gas constant and absolute temperature, respectively. $\psi {\left(x,y,t\right)}^{\mathrm{s}}$ expresses the fluctuation of the chemical potential of the adatoms by the change in surface height, i.e.,

$$\psi {\left(x,y,t\right)}^{\mathrm{s}}=-{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }{{\nabla }_{\perp }}^2\zeta {\left(x,y,t\right)}^{\mathrm{s}}$$

(26)

where ${{\nabla }_{\perp }}^2\equiv {\partial }^2/\partial x^2+{\partial }^2/\partial y^2$, ${\mathsf{\Omega }}_{\mathrm{m}}$ is the molar volume of the depositing metal, ${\gamma }^{\ast }$ is the isotropic surface free energy, and $\zeta {\left(x,y,t\right)}^{\mathrm{s}}$ is the surface height fluctuation of the deposit surface. The flux density of the metallic ion at the interface is given as follows:

$$\overrightarrow{n}·{\overrightarrow{j}}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}^{\mathrm{s}}=D_{\mathrm{m}}\left\{\overrightarrow{n}·\nabla c_{\mathrm{m}}{\left(x,y,z,t\right)}^{\mathrm{s}}\right\}$$

(27)

where $\nabla$ indicates $\left(\partial /\partial x,\partial /\partial y,\partial /\partial z\right)$. The sign of the flux density is defined as a plus sign in the case of deposition.

Finally, the surface deformation rate is determined by the deposition rate and mass flux density of the adatom as follows:

$$\frac{\partial }{\partial t}\zeta {\left(x,y,t\right)}^{\mathrm{s}}={\mathsf{\Omega }}_{\mathrm{m}}\left\{\frac{\partial }{\partial t}{c}_{\mathrm{a}\mathrm{d}}{\left(x,y,t\right)}^{\mathrm{s}}+\overrightarrow{n}·{\overrightarrow{j}}_{\mathrm{i}\mathrm{n}\mathrm{c}}^{\mathrm{s}}\right\}$$

(28)

The substitution of Equations (25) to (28) into Equation (24) leads to the surface–height equation.

$$\frac{1}{{\mathsf{\Omega }}_{\mathrm{m}}}\frac{\partial }{\partial t}\zeta {\left(x,y,t\right)}^{\mathrm{s}}=-{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{{\nabla }_{\perp }}^4\zeta {\left(x,y,t\right)}^{\mathrm{s}}+D_{\mathrm{m}}\left\{\overrightarrow{n}·\nabla c_{\mathrm{m}}{\left(x,y,z,t\right)}^{\mathrm{s}}\right\}$$

(29)

where ${{\nabla }_{\perp }}^4$ represents ${\left({\partial }^2/\partial x^2+{\partial }^2/\partial y^2\right)}^2$.

2.2.2. Electrochemical Boundary Conditions

As has been discussed above, the electron transfer is assumed to be in quasi-equilibrium.

$$\mathrm{M}\left(\mathrm{a}\mathrm{d}\right)\leftrightarrows {\mathrm{M}}^{{\mathrm{z}}_{\mathrm{m}}}\left(\mathrm{I}\mathrm{H}\mathrm{P}\right)+z_{\mathrm{m}}{\mathrm{e}}^{-}\left(\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}\right)$$

(30)

where $\mathrm{M}\left(\mathrm{a}\mathrm{d}\right)$ and ${\mathrm{M}}^{{\mathrm{z}}_{\mathrm{m}}}\left(\mathrm{I}\mathrm{H}\mathrm{P}\right)$ are the adatom and the metallic ion at the inner Helmholtz plane (IHP), respectively. ${\mathrm{e}}^{-}\left(\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}\right)$ is a free electron in the electrode, and $z_{\mathrm{m}}$ is the charge number of the metallic ion.

According to Equation (30), their chemical and electrochemical potentials are related to the following:

$${\mu }_{\mathrm{a}\mathrm{d}}\left(x,y,t\right)={\overline{\mu }}_{\mathrm{m}}\left(x,y,{\zeta }^s,t\right)+{\mathrm{z}}_{\mathrm{m}}{\overline{\mu }}_{\mathrm{e}}\left(x,y,t\right)$$

(31)

where ${\mu }_{\mathrm{a}\mathrm{d}}$ is the chemical potential of the adatoms, ${\overline{\mu }}_{\mathrm{m}}$ is the electrochemical potential of the metallic ions, and ${\overline{\mu }}_{\mathrm{e}}$ is the electrochemical potential of the electrons. ${\zeta }^s$ denotes $\zeta {\left(x,y,t\right)}^{\mathrm{s}}$, i.e., the surface height fluctuation via 3D nucleation.

Here, if a minute projection by the electrodeposition accidentally happens in the diffusion layer, the concentration of the metallic ions fluctuates around it, as expressed by Equation (1). The fluctuation of the free electrons in the metal phase is neglected due to their large mobility in light speed, so the fluctuation of the chemical potential of the adatoms is expressed by the fluctuation of the electrochemical potential of the metallic ions.

$$\mathsf{\delta }{{\mu }_{\mathrm{a}\mathrm{d}}\left(x,y,t\right)}^{\mathrm{s}}=\mathsf{\delta }{{\overline{\mu }}_{\mathrm{m}}\left(x,y,{\zeta }^{\mathrm{s}},t\right)}^{\mathrm{s}}$$

(32)

At the top of the projection, the chemical potential of the adatoms also fluctuates according to Equation (26):

$$\mathsf{\delta }{{\mu }_{\mathrm{a}\mathrm{d}}\left(x,y,t\right)}^{\mathrm{s}}=\psi {\left(x,y,t\right)}^{\mathrm{s}}$$

(33)

On the other hand, the fluctuation of the electrochemical potential of the metallic ions induces the fluctuation of the concentration overpotential arising from the projection.

$$\mathsf{\delta }{{\overline{\mu }}_{\mathrm{m}}\left(x,y,{\zeta }^{\mathrm{s}},t\right)}^{\mathrm{s}}=z_{\mathrm{m}}F\mathsf{\delta }\mathsf{H}{\left(x,y,{\zeta }^{\mathrm{s}},t\right)}^{\mathrm{s}}$$

(34)

where F is the Faraday constant, and ${\mathsf{\delta }\mathrm{H}}^{\mathrm{s}}$ denotes the fluctuation of the concentration overpotential, expressed by the following Nernst-type equation:

$$\mathsf{H}\left(x,y,z,t\right)=\frac{RT}{z_{\mathrm{m}}F}\mathrm{l}\mathrm{n}\left\{\frac{C_{\mathrm{m}}\left(x,y,z,t\right)}{C_{\mathrm{m}}^{\ast }\left(z=\infty \right)}\right\}$$

(35)

where $C_{\mathrm{m}}^{\ast }\left(z=\infty \right)$ is the bulk concentration.

The fluctuation is explicitly written as follows:

$$\mathsf{\delta }\mathsf{H}{\left(x,y,{\zeta }^s,t\right)}^{\mathrm{s}}=\frac{RT}{z_{\mathrm{m}}F}\frac{{c_{\mathrm{m}}\left(x,y,{\zeta }^{\mathrm{s}},t\right)}^{\mathrm{s}}}{〈C_{\mathrm{m}}\left(x,y,0,t\right)〉}$$

(36)

where $〈C_{\mathrm{m}}\left(x,y,0,t\right)〉$ represents the average surface concentration equal to $C_{\mathrm{m}}^{\ast }\left(z=0\right)$, where the asterisk means electrostatic equilibrium. To derive Equation (36), we assumed the condition $\left|{c_{\mathrm{m}}\left(x,y,{\zeta }^{\mathrm{s}},t\right)}^{\mathrm{s}}\right|\ll 〈C_{\mathrm{m}}\left(x,y,0,t\right)〉.$ The concentration fluctuation at the top of the projection is expanded in the first order for ${\zeta }^{\mathrm{s}}$.

$${c_{\mathrm{m}}\left(x,y,{\zeta }^{\mathrm{s}},t\right)}^{\mathrm{s}}={c_{\mathrm{m}}\left(x,y,0,t\right)}^{\mathrm{s}}+L_{\mathrm{m}}\zeta {\left(x,y,t\right)}^{\mathrm{s}}$$

(37a)

where $L_{\mathrm{m}}$ represents the average concentration gradient in the diffusion layer.

$$L_{\mathrm{m}}\equiv \frac{{\theta }_{\infty }^{\ast }}{〈{\delta }_{\mathrm{c}}〉} (>0)$$

(37b)

where ${\theta }_{\infty }^{\ast }$ is the concentration difference between the bulk and surface, i.e.,

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

(37c)

and $〈{\delta }_{\mathrm{c}}〉$ is the average diffusion layer thickness.

Substituting Equations (26) and (33) into Equation (32), and inserting Equations (34), (36), and (37a) into the resulting equation, we obtain the electrochemical boundary condition for the symmetrical fluctuations.

$${c_{\mathrm{m}}\left(x,y,0,t\right)}^{\mathrm{s}}=-L_{\mathrm{m}}\zeta {\left(x,y,t\right)}^{\mathrm{s}}-\frac{{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }}{RT}〈C_{\mathrm{m}}\left(x,y,0,t\right)〉{{\nabla }_{\perp }}^2\zeta {\left(x,y,t\right)}^{\mathrm{s}}$$

(38)

2.2.3. Characteristic Equations of the Nano-MHD Vortexes and 3D Nuclei

Equations (29) and (38) are transformed via Fourier transformation concerning x- and y-coordinates into the following amplitude equations:

$$\left(\frac{\mathrm{d}}{\mathrm{d}t}+{\mathsf{\Omega }}_{\mathrm{m}}^2{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{k}^4\right){\mathsf{{\rm Z}}}_{\mathrm{j}}^0{\left(t\right)}^{\mathrm{s}}={\mathsf{\Omega }}_{\mathrm{m}}{D}_{\mathrm{m}}\mathrm{D}{\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}$$

(39a)

and

$$\left(L_{\mathrm{m}}-\frac{{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }}{RT}〈C_{\mathrm{m}}\left(x,y,0,t\right)〉k^2\right){\mathsf{{\rm Z}}}_{\mathrm{j}}^0{\left(t\right)}^{\mathrm{s}}=-{\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}$$

(39b)

where ${\mathsf{{\rm Z}}}_{\mathrm{j}}^0{\left(t\right)}^{\mathrm{s}}$ and ${\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}$ denote the amplitudes of $\zeta {\left(x,y,t\right)}^{\mathrm{s}}$ and ${c_{\mathrm{m}}\left(x,y,0,t\right)}^{\mathrm{s}}$ on the rigid surface (j = r) and the free surface (j = f), respectively, and $\mathrm{D}{\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}$ indicates ${\left\{\left(\mathrm{d}/\mathrm{d}\mathrm{z}\right){\mathsf{\Theta }}_{\mathrm{j}}^0{\left(z,t\right)}^{\mathrm{s}}\right\}}_{z=0}$. k denotes the wavenumber defined by ${\left(k_{\mathrm{x}}^2+k_{\mathrm{y}}^2\right)}^{1/2},$ and $k_{\mathrm{x}}$ and $k_y$ are the x- and y-components of the wavenumber, respectively.

Substituting ${\mathsf{{\rm Z}}}_{\mathrm{j}}^0{\left(t\right)}^{\mathrm{s}}$ from Equation (39b) into Equation (39a), we obtain the amplitude equation of the concentration fluctuation:

$$\left(\frac{\mathrm{d}}{\mathrm{d}t}+{\mathsf{\Omega }}_{\mathrm{m}}^2{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{k}^4\right){\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}{=-\mathsf{\Omega }}_{\mathrm{m}}{D}_{\mathrm{m}}\left(L_{\mathrm{m}}-\frac{{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }}{RT}〈C_{\mathrm{m}}\left(x,y,0,t\right)〉k^2\right)\mathrm{D}{\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{o}\mathrm{r} \mathrm{f}$$

(40a)

Here, using the representative length, $d^{\mathrm{s}}$, of 3D nucleation in the second generation, we introduce the nondimensional wavenumber, $a^{\mathrm{s}}$, as well as the nondimensional coordinates x, y, and z. Equation (40a) is therefore revised as follows:

$$\begin{array}{c}\left(\frac{\mathrm{d}}{\mathrm{d}t}+{{\mathsf{\Omega }}_{\mathrm{m}}}^2{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{{d^{\mathrm{s}}}^{4}RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{a^{\mathrm{s}}}^4\right){\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}={-{d^{\mathrm{s}}}^{-1}\mathsf{\Omega }}_{\mathrm{m}}{D}_{\mathrm{m}}\left(L_{\mathrm{m}}-\frac{{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }}{{d^{\mathrm{s}}}^{2}RT}〈C_{\mathrm{m}}\left(x,y,0,t\right)〉{a^{\mathrm{s}}}^2\right)\mathrm{D}{\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}\\ \hfill \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}\end{array}$$

(40b)

where the nondimensional parameters are defined by $a^{\mathrm{s}}\equiv k{d}^{\mathrm{s}}, \mathrm{D}\equiv \mathrm{D}{d}^{\mathrm{s}}$, $x\equiv x/d^{\mathrm{s}}$ and $y\equiv y/d^{\mathrm{s}}$. In the following, introducing the vorticity coefficient functions, VCFs, of the rigid- and free-surface nano-MHD vortexes, we obtain the characteristic equations of rigid- and free-surface nucleation with nano-MHD vortexes in the second generation.

As was shown initially, since a symmetrical fluctuation is essentially one among the asymmetrical fluctuations, from the general amplitude equation of the asymmetrical concentration fluctuation based on the two-layer system formulated in Part 1 [1], we can derive the amplitude equation of the symmetrical concentration fluctuation on the rigid and free surfaces as follows:

$${\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}=-{\mathrm{f}}_{\mathrm{j}}^s{\left(a^{\mathrm{s}}\right)}^{-1}\frac{2a^{\mathrm{s}}{\beta }_{\mathrm{i}}^{\mathrm{s}}}{z_{\mathrm{m}}F{D}_{\mathrm{m}}{S}^{\ast \mathrm{s}}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r}, \mathrm{i}=1 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{j}=\mathrm{f}, \mathrm{i}=0$$

(41a)

where ${\beta }_{\mathrm{i}}^{\mathrm{s}}$ is the vorticity coefficient of the rigid-surface (i = 1) and free-surface (i = 0) vortexes, and ${\mathrm{f}}_{\mathrm{j}}^s\left(a^{\mathrm{s}}\right)$ is the vorticity coefficient function, VCF, of the rigid-surface (j = r) and free-surface (j = f) nano-MHD vortexes. In Part 1 [1], they were generally used in the equations of the amplitudes, ${\mathsf{\Omega }}_{\mathrm{j}}^0\left(z,t\right)$, of the z-components of the vorticity fluctuations, ${\omega }_{\mathrm{z},\mathrm{j}}$, i.e., ${\mathsf{\Omega }}_{\mathrm{r}}^0\left(z,t\right)={\beta }_{1}z{\mathrm{e}}^{az}$ and ${\mathsf{\Omega }}_{\mathrm{f}}^0\left(z,t\right)={\beta }_0\left(1-az\right){\mathrm{e}}^{az}$. From Part 1, the gradient of the amplitude of the symmetrical concentration fluctuation is also obtained via the following:

$${\mathrm{D}\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}=\frac{2a^{\mathrm{s}}{\beta }_{\mathrm{i}}^{\mathrm{s}}}{z_{\mathrm{m}}F{D}_{\mathrm{m}}{s}^{\ast \mathrm{s}}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r}(\mathrm{i}=1) \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}(\mathrm{i}=0)$$

(41b)

Substituting Equations (41a) and (41b) into Equation (40b), we have the characteristic equation of the vorticity coefficient, ${\beta }_{\mathrm{i}}^{\mathrm{s}}$:

$$\frac{\mathrm{d}}{\mathrm{d}t}{\beta }_{\mathrm{i}}^{\mathrm{s}}=\left\{{{d^{\mathrm{s}}}^{-1}\mathsf{\Omega }}_{\mathrm{m}}{D}_{\mathrm{m}}\left(L_{\mathrm{m}}-\frac{{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }}{{d^{\mathrm{s}}}^{2}RT}〈C_{\mathrm{m}}\left(x,y,0,t\right)〉{a^{\mathrm{s}}}^2\right)\right.{\mathrm{f}}_{\mathrm{i}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)-{{\mathsf{\Omega }}_{\mathrm{m}}}^2\left.{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{{d^{\mathrm{s}}}^{4}RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{a^{\mathrm{s}}}^4\right\}{\beta }_{\mathrm{i}}^{\mathrm{s}}$$

(42a)

Here, as shown in Equations (22a) and (22b), the VCFs converge to ${\mathrm{f}}_{\mathrm{i}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)=a^{\mathrm{s}}$ with decreasing representative length, $d^{\mathrm{s}}$. This result indicates that the activated MHD vortexes vanish as $d^{\mathrm{s}}$ decreases with the lower generation.

When the average surface concentration $〈C_{\mathrm{m}}\left(x,y,0,t\right)〉$ is regarded as constant, i.e., under a potentiostatic condition, Equation (42a) is solved as follows:

$${\beta }_{\mathrm{i}}^{\mathrm{s}}\left(t\right)={\beta }_{\mathrm{i}}^{\mathrm{s}}\left(0\right)\mathrm{e}\mathrm{x}\mathrm{p}\left\{p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)t\right\} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}$$

(42b)

where $p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$ denotes the amplitude factor of the rigid- and free-surface vortexes. As will be shown later, if the amplitude factor, $p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$, is positive for some of the wavenumber $a^{\mathrm{s}}$, the vortex coefficient, ${\beta }_{\mathrm{i}}^{\mathrm{s}}\left(t\right)$, increasing with time gives rise to unstably growing nano-MHD vortexes. However, when $p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$ is kept negative for all $a^{\mathrm{s}}$ values, all of the ${\beta }_{\mathrm{i}}^{\mathrm{s}}\left(t\right)$ values decreasing with time lead to stably dwindling vortexes. The instability of the nano-MHD vortexes depends on the sign of the amplitude factor, $p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right).$

A similar relationship to that in Equation (42a) concerning the surface height fluctuation is also obtained; from Equations (41a) and (41b), we have the following:

$${\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}}=-{\mathrm{f}}_{\mathrm{j}}^s{\left(a^{\mathrm{s}}\right)}^{-1}\mathrm{D}{\mathsf{\Theta }}_{\mathrm{j}}^0{\left(0,t\right)}^{\mathrm{s}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}$$

(43)

Substituting Equation (43) into Equations (39a) and (39b), and adding the resulting equations together, we obtain the amplitude equation of the surface height fluctuation:

$$\frac{\mathrm{d}}{\mathrm{d}t}{\mathsf{{\rm Z}}}_{\mathrm{j}}^0{\left(t\right)}^{\mathrm{s}}=\left\{{{d^{\mathrm{s}}}^{-1}\mathsf{\Omega }}_{\mathrm{m}}{D}_{\mathrm{m}}\left(L_{\mathrm{m}}-\frac{{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }}{{d^{\mathrm{s}}}^{2}RT}〈C_{\mathrm{m}}\left(x,y,0,t\right)〉{a^{\mathrm{s}}}^2\right)\right.{\mathrm{f}}_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)-\left.{{\mathsf{\Omega }}_{\mathrm{m}}}^2{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{{d^{\mathrm{s}}}^{4}RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{a^{\mathrm{s}}}^4\right\}{\mathsf{{\rm Z}}}_{\mathrm{j}}^0{\left(t\right)}^{\mathrm{s}}$$

(44a)

Equation (44a) is similar to Equation (42a), so we also obtain the following equation:

$${\mathsf{{\rm Z}}}_{\mathrm{j}}^{\mathrm{s}}\left(t\right)={\mathsf{{\rm Z}}}_{\mathrm{j}}^{\mathrm{s}}\left(0\right)\mathrm{e}\mathrm{x}\mathrm{p}\left\{p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)t\right\} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}$$

(44b)

Namely, the surface height fluctuation is also controlled by the same amplitude factor, $p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$, which is defined by the following:

$$p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)\equiv {{d^{\mathrm{s}}}^{-1}\mathsf{\Omega }}_{\mathrm{m}}{D}_{\mathrm{m}}\left(L_{\mathrm{m}}-\frac{{\mathsf{\Omega }}_{\mathrm{m}}{\gamma }^{\ast }}{{d^{\mathrm{s}}}^{2}RT}〈C_{\mathrm{m}}\left(x,y,0,t\right)〉{a^{\mathrm{s}}}^2\right){\mathrm{f}}_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)-{{\mathsf{\Omega }}_{\mathrm{m}}}^2{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{{d^{\mathrm{s}}}^{4}RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{a^{\mathrm{s}}}^4 \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}$$

(45a)

Under limiting diffusion, i.e., $〈C_{\mathrm{m}}\left(x,y,0,t\right)〉\approx 0$, Equation (45a) is simplified via the following:

$$p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)={{d^{\mathrm{s}}}^{-1}\mathsf{\Omega }}_{\mathrm{m}}{D}_{\mathrm{m}}{L}_{\mathrm{m}}{\mathrm{f}}_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)-{{\mathsf{\Omega }}_{\mathrm{m}}}^2{\gamma }^{\ast }\frac{D_{\mathrm{a}\mathrm{d}}}{{d^{\mathrm{s}}}^{4}RT}{C}_{\mathrm{a}\mathrm{d}}^{\ast }{a^{\mathrm{s}}}^4$$

(45b)

The first term on the right-hand side of Equation (45b) is the main part of the amplitude factor, and the second term is the additional one expressing the suppression by surface diffusion [76]. Therefore, the sign of $p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$ is mainly determined by the first term. As mentioned above, inserting the stationary-solution condition, ${\mathrm{f}}_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)=a^{\mathrm{s}}$, in Equations (22a) and (22b) into Equations (45a) and (45b), we can make the amplitude factor exist in a stationary solution [69,70,71]. In the case of cathodic deposition, the concentration gradient, $L_{\mathrm{m}}$, is positive and considered to consist of the main part, so $p_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$ becomes positive for a certain wavenumber range. Namely, 3D nucleation in a stationary solution is always unstable. On the other hand, the vorticity coefficient function, VCF, ${\mathrm{f}}_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$, indicates the contribution of the nano-MHD vortexes to 3D nucleation. As shown in Figure 6, it takes positive and negative values for right- and left-handed systems of the upper generation, respectively, so the MHD vortexes sometimes promote and sometimes suppress the instability. This behavior of MHD vortexes is different from the behavior under a parallel magnetic field; the MHD vortexes under a parallel magnetic field only give rise to suppression [72].

Therefore, in general, the instability of the nucleation under MHD vortexes is determined by the sign of the product of the main part of the amplitude factor in a stationary solution of the VCF, i.e., positive and negative correspond to unstable and stable for the MHD vortexes and nucleation in each generation. This process is simply expressed in a schematic form:

$$\left[\mathrm{N}\mathrm{I} \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r} \mathrm{M}\mathrm{H}\mathrm{D}\mathrm{V}\mathrm{s}\right]=\left[\mathrm{N}\mathrm{I} \mathrm{i}\mathrm{n} \mathrm{a} \mathrm{S}.\mathrm{S}.\right]\times \left[\mathrm{V}\mathrm{C}\mathrm{F} \right]$$

(46)

where NI denotes nucleation instability, MHDV represents the MHD vortex, and S.S. represents a stationary solution. “[ ]” expresses the sign of each term. It should be noted that Equation (46) also expresses the instability of the MHD vortexes themselves.

3. Results and Discussion

As was shown in Part 1 [1], in the first generation, the micro-MHD vortexes unstably develop with 2D nuclei under a VMHDF. Since the rotation of the VMHDF forms a right-handed system with the external magnetic field, $B_0\tilde{\mathsf{\Omega }}>0$, as shown in Equation (21a), the VCFs of the micro-MHD vortexes are defined as positive functions, ${\mathrm{f}}_{\mathrm{j}}^{\mathrm{a}}\left(a^{\mathrm{a}}\right)>0$ for j = r and f. In the absence of chloride ions, 2D nucleation is unstable in a stationary solution [65], so the main part of the amplitude factor in a stationary solution is positive. BY Equation (46), the instability of 2D nucleation under micro-MHD vortexes is therefore defined by the sign of the product of the stationary-solution main part of the VCF, i.e., due to the derived positive values, the micro-MHD vortexes unstably develop together with 2D nuclei. Namely, the instability of 2D nucleation in a stationary solution is promoted by the micro-MHD vortexes.

Since the value, as well as the sign of the amplitude factor under MHD vortexes is approximately determined by the same product, the difference in the instability strength between the rigid-surface and free-surface vortexes results from the values of the corresponding VCFs. As has been discussed in Equation (21c), they generally have the relationship $\left|{\mathrm{f}}_{\mathrm{r}}^{\mathrm{i}}\left(a\right)\right|<\left|{\mathrm{f}}_{\mathrm{f}}^{\mathrm{i}}\left(a\right)\right|$ in every generation “i”, i.e., the absolute value of the free-surface VCF is larger than that of the rigid-surface VCF. This means that in the unstable case, the positive free-surface amplitude factor is larger than the positive rigid-surface amplitude factor. Namely, the free-surface micro-MHD vortexes grow faster than the rigid-surface ones, so the free-surface ones preferentially receive the precession from the upper generation vortex, VMHDF, rotating in the same direction.

As mentioned initially, chiral nucleation occurs only on the rigid surfaces because the free surfaces rotating with the vortexes yield no chirality. Then, to conserve a local angular momentum, as shown in Figure 3d, a pair of rigid- and free-surface vortexes rotate oppositely. Therefore, following the precessional rotation of the free-surface micro-MHD vortexes, the rigid-surface micro-MHD vortexes rotate in the opposite direction, forming a left-handed system, $B_0\tilde{\mathsf{\Omega }}<0$, with the external magnetic field. As a result, the first-generation 2D nuclei under the rigid-surface micro-vortexes are created with opposite chirality to that of the VMHDF.

At the same time, the left-handed system of a rigid-surface micro-MHD vortex leads to negative VCFs of the nano-MHD vortexes in the second generation, i.e., ${\mathrm{f}}_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$ < 0 for j = r and f in Equation (21b). As discussed above, 3D nucleation in a stationary solution is always unstable, i.e., the main part of the amplitude factor, the concentration gradient $L_{\mathrm{m}}>0$ in Equation (45b), is kept positive so that the products of the main part of the VCFs, i.e., the main parts of the amplitude factors under nano-MHD vortexes, become negative, $L_{\mathrm{m}}{\mathrm{f}}_{\mathrm{j}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)$ < 0. This time, the unstable 3D nucleation in a stationary solution is stabilized by the nano-MHD vortexes. From Equation (21c), the relationship ${\mathrm{f}}_{\mathrm{f}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)<{\mathrm{f}}_{\mathrm{r}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)<0$ is derived, so concerning the main parts of the nano-MHD vortexes under the rigid-surface micro-MHD vortexes, the relationship, ${L_{\mathrm{m}}\mathrm{f}}_{\mathrm{f}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)<{L_{\mathrm{m}}\mathrm{f}}_{\mathrm{r}}^{\mathrm{s}}\left(a^{\mathrm{s}}\right)<0$ is derived. According to Equations (42b) and (45b), the rigid-surface nano-MHD vortexes dwindle more slowly than the free-surface ones. Therefore, the precession from the rigid-surface micro-MHD vortex directly transfers to the rigid-surface nano-MHD vortexes, forming a left-handed system again in the third generation. From these discussions, it is concluded that in stable nucleation, lower-generation rigid-surface vortexes obtain the same chirality as upper-generation rigid-surface vortexes, whereas they receive the opposite chirality in the unstable case. Figure 8 exhibits the amplitude factors of 3D nucleation in a stationary solution and the amplitude factors under nano-MHD vortexes in the right-handed and left-handed systems. Though the left-handed system is derived from the case without chloride ions, as will be discussed later, the right-handed system corresponds to the case with chloride ions. As shown in Figure 8b,c, the former and latter lead to stable and unstable 3D nucleation, changing chirality.

In the third generation, though the detailed mechanism has been not yet established, screw dislocations were created by some instability based on the mass transfer and electrode reaction, which we could discuss in the same way: In the absence of chloride ions, because the screw dislocation process in a stationary solution is unstable [77], the main part of its amplitude factor must be positive. However, due to the left-handed system of the rigid-surface nano-MHD vortex, the VCFs of the ultra-micro-MHD vortexes forming the screw dislocations become negative functions. As a result, the main part of the amplitude factors of the screw dislocations under the ultra-micro-MHD vortexes are determined as negative, i.e., they are stable, so the rigid-surface ultra-micro-MHD vortexes show the same chirality of the left-handed system as that of the rigid-surface nano-MHD vortexes. This result represents that, as shown in the experimental results obtained by Mogi and coworkers [8,9,10,11,12,13,14,15,16,17,18,19,20,21], the CW activity (D-activity) and ACW activity (L-activity) of the deposit surface appear under antiparallel (upward) and parallel (downward) magnetic fields, respectively.

On the contrary, in the presence of chloride ions, as has been shown in Part 1 [1], due to the specific adsorption of chloride ions in the electric double layer, the micro-MHD vortexes and 2D nucleation turn stable, so the rigid-surface micro-MHD vortexes in the first generation directly progress toward the chirality of the VMHDF, i.e., the rotation of the right-handed system of the VMHDF with the external magnetic field is inherited in the first generation. The VCFs of the micro-MHD vortexes therefore become positive functions in the second generation. However, as mentioned above, whether chloride ions are present or not, 3D nucleation in a stationary solution is always unstable, so the main part of the amplitude factor in a stationary solution is kept positive. This time, due to the positive products of the positive main part of the positive VCFs, the nano-MHD vortexes and 3D nucleation become unstable, so the rigid-surface nano-MHD vortexes of the second generation obtain the opposite chirality to that of the rigid-surface micro-MHD vortexes of the first generation, i.e., the left-handed-system’s chirality. Therefore, the VCFs of the ultra-micro vortexes of the third generation become negative functions. Here, as for screw dislocation, chloride ions are well-known leveling reagents for copper deposition, which make copper screw dislocations vanish and become stabilized in a stationary solution [77]. This implies that the main part of the amplitude factor of screw dislocation in a stationary solution takes a negative value. The positive products of the negative main part of the negative VCFs lead to the instability of screw dislocation and ultra-micro MHD vortexes. At the same time, this instability gives rise to the change in chirality from the left-handed system to that of the right-handed system; the chiral activity of screw dislocations is reversed via the addition of chloride ions.

In the three-generation model without chloride ions, the chiral activity of screw dislocations in the third generation is therefore consistent with that of the rigid-surface 2D nucleus shown in the first generation: For a positive magnetic field, $B_0>0$, i.e., for an upward antiparallel magnetic field, ACW (VMHDF)$\to$ CW (2D nucleus) $\to$ CW (3D nucleus) $\to$ CW (screw dislocation). For a negative magnetic field, i.e., for a downward parallel magnetic field, $B_0<0$, CW (VMHDF)$\to \mathrm{A}$CW (2D nucleus) $\to$ ACW (3D nucleus) $\to$ ACW (screw dislocation). Namely, under an upward (anti-parallel) magnetic field $B_0>0$, D-activity emerges, whereas under a downward (parallel) magnetic field $B_0<0$, screw dislocations with L-activity are produced. In the presence of chloride ions, the chiral activity of screw dislocations in the third generation is changed to reverse one: For an upward antiparallel (positive) magnetic field, $B_0>0$, ACW (VMHDF) $\to$ ACW (2D nucleus) $\to$ CW (3D nucleus) $\to$ ACW (screw dislocation). For a downward parallel (negative) magnetic field, $B_0<0$, CW (VMHDF) $\to$ CW (2D nucleus) $\to$ ACW (3D nucleus) $\to$ CW (screw dislocation). In the presence of chloride ions, due to specific adsorption on the electric double layer, we can expect that inversed chirality appears, depending on the direction of the magnetic field, i.e., for $B_0>0$ and $B_0<0$, L-activity and D-activity arise, respectively.

The above discussions are summarized as follows:

(1)

The rigid-surface vortex rotation, $\tilde{\mathsf{\Omega }}\left(\mathrm{r}\mathrm{i}\mathrm{g}\right)\left(\mathrm{i}\right)$, of the upper generation, “i” and the external magnetic field, $B_0$ forms a right-handed system, $B_0\tilde{\mathsf{\Omega }}\left(\mathrm{r}\mathrm{i}\mathrm{g}\right)\left(\mathrm{i}\right)>$, 0 or a left-handed system, $B_0\tilde{\mathsf{\Omega }}\left(\mathrm{r}\mathrm{i}\mathrm{g}\right)\left(\mathrm{i}\right)<$ 0. The vorticity coefficient function, $\mathrm{V}\mathrm{C}\mathrm{F}\left(\mathrm{i}+1\right)$, of the lower generation, “i+1”, is defined as a positive function,$\mathrm{V}\mathrm{C}\mathrm{F}\left(\mathrm{i}+1\right)>$ 0, for $B_0\tilde{\mathsf{\Omega }}\left(\mathrm{r}\mathrm{i}\mathrm{g}\right)\left(\mathrm{i}\right)>$ 0 and a negative function, $\mathrm{V}\mathrm{C}\mathrm{F}\left(\mathrm{i}+1\right)<0$, for $B_0\tilde{\mathsf{\Omega }}\left(\mathrm{r}\mathrm{i}\mathrm{g}\right)\left(\mathrm{i}\right)<$ 0.

(2)

The main part (MP) of the amplitude factor in a stationary solution (S.S.) in the lower generation, $\mathrm{M}\mathrm{P}(\mathrm{S}.\mathrm{S}.)\left(\mathrm{i}+1\right)$, is defined as positive, $\mathrm{M}\mathrm{P}(\mathrm{S}.\mathrm{S}.)\left(\mathrm{i}+1\right)>0$ or negative, $\mathrm{M}\mathrm{P}(\mathrm{S}.\mathrm{S}.)\left(\mathrm{i}+1\right)<0$, depending on whether the nucleation in a stationary solution is “unstable” or “stable”.

(3)

The nucleation under the rigid-surface vortexes is determined to be unstable for $\mathrm{M}\mathrm{P}(\mathrm{S}.\mathrm{S}.)\left(\mathrm{i}+1\right)·\mathrm{V}\mathrm{C}\mathrm{F}\left(\mathrm{i}+1\right)>0$ and stable for $\mathrm{M}\mathrm{P}(\mathrm{S}.\mathrm{S}.)\left(\mathrm{i}+1\right)·\mathrm{V}\mathrm{C}\mathrm{F}\left(\mathrm{i}+1\right)<0$.

(4)

For the unstable case, $\mathrm{M}\mathrm{P}(\mathrm{S}.\mathrm{S}.)\left(\mathrm{i}+1\right)·\mathrm{V}\mathrm{C}\mathrm{F}\left(\mathrm{i}+1\right)>0$, the chirality of the lower generation, ‘i+1”, is changed from that of the upper generation, “i”, and for the stable case, $\mathrm{M}\mathrm{P}(\mathrm{S}.\mathrm{S}.)\left(\mathrm{i}+1\right)·\mathrm{V}\mathrm{C}\mathrm{F}\left(\mathrm{i}+1\right)<0$, the chirality of the present lower generation, “i + 1”, preserves that of the upper generation, “i”. In Figure 9, we exhibit the flow chart for MHD vortexes to obtain chirality in the three generations and the actual chirality obtained in each generation in the absence and presence of chloride ions.

The experimental result of copper deposition onto a copper electrode with a larger diameter of 3 mm in a copper sulfate solution from a study by Mogi and coworkers told us that the reversal of chirality by adding chloride ions is partial, i.e., only in the upward magnetic field does D-activity change to L-activity, whereas L-activity in the downward magnetic field keeps the same chirality [14,20,21]. This result strongly suggests disturbance from magneto-convection by a gradient field force, which is induced from the magnetic field gradient applied to a paramagnetic material, being dependent on the magnetic field direction. A copper sulfate solution is paramagnetic, and it is well known that the characteristic magneto-convection arises through a vertical magnetic field gradient [54].

Finally, following Appendix A, the surface morphology of 3D nuclei in the second generation was calculated. Figure 10a shows a calculated 3D nucleus formed on a rigid electrode surface under an antiparallel upward magnetic field $\left(B_0>0\right)$. Under a stable CW nano-MHD vortex, though not the third generation, a screw dislocation-like 3D nucleus with chirality is formed. The clear-cut surfaces without obscurity due to diffusion come from the phase cutting of the symmetrical fluctuation shown in Figure 4. This result suggests that the chiral screw dislocations of the third generation also evolve from the formation process, similar to the chiral 3D nuclei of the second generation. At this time, a question is brought about: how should we determine the chirality of it? Namely, when climbing up the slope shown in Figure 10a, we turn anticlockwise. However, we must turn clockwise to go down it. The nucleus grows under a rigid-surface nano-MHD vortex, which has, as shown in Figure 3c, a downward flow. An enantiomeric reagent also approaches downwards, so the chirality should be defined as a system with a downward direction, i.e., for the present case, CW chirality and D-activity are derived. Figure 10b is a deposit surface of the second generation after 100 rounds of nucleation. In every nucleation, chiral 3D nuclei are formed at fixed points on the rigid surfaces, whereas on the free surfaces covered with ionic vacancies, due to rotation with the micro-MHD vortexes, nuclei without chirality are developed.

4. Conclusions

In the three-generation model, chirality transfers from the upper generation the to lower generation via the instability or stability difference between the rigid- and free-surface MHD vortexes rotating in opposite directions. Since the effective viscosity increases with lowering generations, for the vortexes to rotate in the second and third generations, the kinematic viscosity must decrease to a zero level. Ionic vacancies therefore work as atomic-scale lubricants, lowering the viscosity of the vacancy layer on the electrode to almost zero. At the same time, the vortexes promote or suppress 3D nucleation, depending on whether the system is right-handed or left-handed.

The addition of chloride ions reverses the chiral activity of the electrode. However, in the case of copper deposition, due to magneto-convection from the gradient force to the paramagnetic copper sulfate solution, such reversal becomes partial, leading to a breakdown in chiral symmetry.