Solute Concentration Distribution in the Vicinity of Solid-Liquid Interface under the Imposition of a Time-Varying Force

Abstract

Mass transfer is often the rate determining step for solid-liquid reaction, such as an electroplating process in automotive industry and a refining process in metallurgical industry. The decrease of concentration boundary layer thickness through the excitation of convection is adapted to enhance the solid-liquid chemical reaction rate. Therefore, traditional methods excite a macro-scale flow in the bulk liquid. Because the concentration boundary layer exists in the velocity boundary layer, the traditional methods have the limitation in enhancing mass transfer rate. Therefore, a new method was proposed, which imposes force directly near the solid-liquid interface. In the past research, force, with or without an oscillating component was imposed near the solid-liquid interface during the dissolution of a Cu anode into a Cu²⁺ aqueous solution. The increase of Cu²⁺ concentration under the force imposition with oscillating component was suppressed compared to that by imposing the force without oscillating component just above the center of the anode. This research evaluated the dissolved Cu²⁺ concentration distribution and the liquid flow pattern in the whole vicinity of the solid-liquid interface under the force imposition with or without oscillation component. The results indicated that by imposing the force with oscillating component, the increase of the Cu²⁺ concentration was suppressed in the whole vicinity of the solid-liquid interface, and the Cu²⁺ concentration distributed more uniformly near the solid-liquid interface. This might be because of the excitation of circulating micro-scale flows near the side parts of the anode surface.

1. Introduction

For solid-liquid chemical reaction, mass transfer in concentration boundary layer formed in a liquid is often the rate determining step such as an electroplating process in automotive industry and a refining process in metallurgical industry.^1,2) The mass transfer rate in it depends on diffusion and convection. The former is described by the Fick’s first law, and its intensity depends on the diffusion coefficient and the concentration gradient. Decrease of the concentration boundary layer thickness is an effective way to enhance mass transfer. The convection also contributes to the mass transfer enhancement because it decreases the concentration boundary layer thickness^3,4) Therefore, an excitation of a macro-scale flow in the bulk has been used as traditional method to enhance mass transfer.^5,6,7,8,9)

The velocity boundary layer in which flow adjusts from zero velocity at the solid-liquid interface to the maximum in the main flow^10,11) is formed near the solid-liquid interface under the macro-scale flow excitation. Its relative thickness to the concentration boundary layer is described by a dimensionless number, Schmidt number.¹²⁾ Because the Schmidt number is much larger than unity for liquids, the concentration boundary layer exists in the velocity boundary layer. This means that an intense convection in the bulk liquid is required for the mass transfer enhancement by the traditional method. This also results in the degradation of product qualities at the same time. The involvement of slags into a liquid metal for the refining process in metallurgy and the formation of large voids in the deposition layer for the electroplating process has been reported.^13,14) Therefore, flow excitation in the bulk liquid does not effectively enhance the mass transfer. Flow excitation near the solid-liquid interface is required for the concentration boundary layer thickness decrease. Based on this concept, a new method was proposed, which excites a force directly near the solid-liquid interface by imposing an electrical current and a magnetic field simultaneously.¹³⁾ By this way, flow is directly excited near the solid-liquid interface, and the decrease of concentration boundary layer thickness and the enhancement of mass transfer are expected.

Since the flow is excited by superimposing the electrical current and the magnetic field, this method can be applied to processes treating a conductive liquid, like a metal refining process and the electroplating process.^15,16,17,18) Furthermore, because flow near the solid-liquid interface can be controlled by controlling the parameters of the electrical current or the magnetic field, quantitative controlment of the mass transfer is expected.^19,20,21)

In the previous study,¹³⁾ a Cu anodic electrode was dissolved into a Cu²⁺ electrolyte aqueous solution. And the dissolved Cu²⁺ concentration was measured just above the anode center under the imposition of the electromagnetic force with or without an oscillating component. The results indicated that increase of Cu²⁺ concentration just above the middle part of the anode surface was suppressed when the force included the oscillating component. However, the Cu²⁺ concentration distribution in the whole vicinity of the anode surface by imposing the force with or without oscillating component has not been studied. Therefore, in this study, the dissolved Cu²⁺ concentration in the whole vicinity of the anode surface has been evaluated, and the time variation and the uniformity of the Cu²⁺ concentration distribution were discussed. The liquid flow patterns were observed to clarify the mechanism for the Cu²⁺ concentration distribution change.

2. Experimental Method

Figure 1 indicates the longitudinal cross-section of the experimental apparatus. A transparent acrylic vessel with a 20 mm inner length, 4 mm depth and 10 mm height was filled with 0.3 mol/L CuSO₄+ 0.1 mol/L H₂SO₄ aqueous solution. Two parallel Cu electrodes were set in the upper and lower parts of the vessel. The upper one was cathode and the lower one was anode. The left and right sides of the lower Cu electrode were covered by 5 mm length insulators with a height of 180 μm. The coordinate system in this investigation was also defined in Fig. 1. The center of the anode surface in the horizontal direction was set as the axis origin, and the horizontal, vertical and anteroposterior directions were indicated as the x-axis. y-axis, and z-axis respectively.

Fig. 1.

Longitudinal cross-section of experimental apparatus.

The three experimental conditions are listed in Table 1 with their abbreviations. The two current conditions were adapted. One was a 15 mA DC current, and the other was a modulate current composed of a 15 mA DC current and a 2 Hz, 30 mAp-p AC current, with the maximum and minimum intensities of 30 mA and 0 mA respectively. The average current intensities of these two conditions were 15 mA. Only the 15 mA DC current was imposed under the “DC condition”, while a static horizontal magnetic field perpendicular to the current direction as shown in Fig. 1 was superimposed with the DC current or the modulate current. These are called the “DC+MF condition” or the “DC+AC+MF condition”, respectively. Thus, the electromagnetic force without or with oscillating component acted on the solution in the positive x-direction under the DC+MF condition and the DC+AC+MF condition. Because the intensity of the magnetic field near the anode was stronger than that near the cathode, the excitation of a counterclockwise circulating macro-scale flow is expected under the DC+MF and DC+AC+MF conditions.

Table 1. Experimental conditions.

	Experimental condition	DC current intensity (mA)	AC current intensity (mAp-p)	Magnetic flux density near anode (T)	Magnetic flux density near cathode (T)	Force condition
1	DC condition	15	0	0	0	no force
2	DC+MF condition	15	0	0.26	0.24	without oscillation component
3	DC+AC+MF condition	15	30	0.26	0.24	with oscillation component

By imposing the electrical current, a Cu²⁺ concentration boundary layer forms near the solid-liquid interface since the solid Cu anode dissolves as Cu²⁺ into the aqueous solution. Based on the Lambert-Beer’s law, the brightness of the aqueous solution used in this investigation depends on the Cu²⁺ concentration. Therefore, the Cu²⁺ concentration can be evaluated by measuring the liquid brightness based on the following equation.^22,23)   

$$c=\frac{{log}_{10}\left(\frac{I_1}{I_2}\right)}{-\epsilon l}+A$$(1)

where A is a constant, c is the Cu²⁺ concentration, I₁ is the brightness of objective liquid, I₂ is the brightness of a standard liquid, l is the light passing length in the absorbing medium, and ε is the extinction coefficient, respectively.

By using this relation, the Cu²⁺ concentration was evaluated 200 μm above the anode (y = 200 μm). A flat light source was set at the back of the vessel for uniform light incident. To exclude the error caused by the natural light, the experiment was conducted in a dark curtain. The brightness of the aqueous solution was recorded by a video recorder with a frame rate of 50 frames per second and a pixel size of 40 μm × 40 μm.

The liquid velocity was evaluated by using polystyrene tracer particles with a diameter of 80 μm. The x-direction velocity was evaluated in the vertical y range of 160–240 μm. For the expression convenience, the measurement range is expressed as the vertical position of y = 200 μm, which is the average vertical position of the vertical range. The y-direction motion of the tracer particle caused by the density difference between the tracer particle and the aqueous solution resulted in the falling down of the tracer particles, which displaced the tracer particles away from the measurement range and led to the experimental error. The terminal velocity u of the tracer particles caused by the density difference can be calculated by the following equation:^24,25)   

$$u=\frac{d^2\mathrm{\Delta }\rho g}{18\eta }$$(2)

where d is the diameter of the tracer particle, g is the gravitational acceleration, Δρ is the density difference between the liquid and the tracer particle and η is the liquid viscosity, respectively.

Because the terminal velocity was calculated as 31 μm/s, the particle displacement time from the measurement range length of 80 μm was 2.6 seconds. Thus, the particle motion measuring time was less than 2.6 s in this experiment to avoid the particle displacement from the measurement range.

3. Experimental Results and Discussion

3.1. Cu²⁺ Concentration Distribution

Time dependence of the Cu²⁺ concentration distributions at the y position of 200 μm under the DC condition, the DC+MF condition, and the DC+AC+MF condition are shown in Fig. 2. The initial concentration distributions show deviation around the initial value of 0.3 mol/L, which was caused by the light reflection on the uneven anode surface, though the flat light source was used in this experiment. However, the spatial distribution and time dependence of the Cu²⁺ concentration can be evaluated because the deviation was small.

Fig. 2.

Time-dependence of Cu²⁺ concentration at 200 μm above anode under (a) DC condition, (b) DC+MF condition and (c) DC+AC+MF condition.

An asymmetric Cu²⁺ concentration distribution was observed under the DC condition. As shown in Fig. 2(a), near the middle part of the anode, where marked by the solid square, the Cu²⁺ concentration was lower than that near the side parts of the anode. This is caused by the current concentration near the side parts due to the length difference between the anode and the cathode.²⁶⁾ Furthermore, as electrical conductivity of the aqueous solution has positive relation with the Cu²⁺ concentration,^27,28) the higher Cu²⁺ concentration near the side parts of the anode further enhanced the current concentration in these regions at the same time. Therefore, the Cu²⁺ concentration difference between near the side parts and near the middle part increased with time.

The uniform distribution of the Cu²⁺ concentration in the range of around −2 mm < x < 2 mm near the middle part of the anode suggests that the uniform electrical current flowed perpendicular to the anode. In this x range, the Cu²⁺ concentrations at 0 s and 20 s were almost the same, whereas it increased obviously at 40 s and 60 s. This indicates that the Cu²⁺ diffusion distance at 20 s was less than 200 μm while it was close to 200 μm at 40 s and 60 s. On the other hand, the diffusion distance x can be estimated by the following equation in the case of one-dimensional diffusion phenomenon:²⁹⁾   

$$x~\sqrt{D\cdot t_D}$$(3)

where D is diffusion coefficient and t_D is diffusion time, respectively.

Because the diffusion coefficient of Cu²⁺ ion in a 0.3 mol/L Cu²⁺ aqueous solution is about 5.5 × 10⁻¹⁰ m²/s,³⁰⁾ the diffusion distance calculated using Eq. (3) is about 105 μm at 20 s, about 148 μm at 40 s and about 182 μm at 60 s. Thus, the observed results near the middle part of the anode are in accordance with the theoretical evaluation results.

Figure 2(b) indicates the time dependence of the Cu²⁺ concentration distribution under the DC+MF condition. The asymmetrical Cu²⁺ concentration distribution with higher Cu²⁺ concentration near the side parts of the anode surface was observed, as marked by the solid square and the dashed square near the left and right parts respectively. The Cu²⁺ concentration obviously increased with time near the left part of the anode surface as marked by the solid square. In addition, a relative uniform Cu²⁺ concentration distribution was observed near the middle part of the anode surface.

Figure 2(c) shows the time dependence of the Cu²⁺ concentration distribution under the DC+AC+MF condition. Compared with that under the DC+MF condition, the Cu²⁺ concentration distributed more uniformly, and the increase of the Cu²⁺ concentration was suppressed. In addition, higher Cu²⁺ concentration near the side parts of the anode surface was not observed.

Figure 3(a) shows the average Cu²⁺ concentration calculation results at the y position of 200 μm under the DC condition, the DC+MF condition, and the DC+AC+MF condition. The Cu²⁺ concentration increased with time under the DC condition, because the dissolved Cu²⁺ from the anode gradually diffused towards the positive y direction. The average concentration under the DC+MF condition was suppressed compared to that under the DC condition. In contrast, no obvious increase of the average Cu²⁺ concentration with time was observed under the DC+AC+MF condition. Compared with that under the DC+MF condition, the average Cu²⁺ concentration was further suppressed.

Fig. 3.

(a) Average Cu²⁺ concentration and (b) standard deviation of Cu²⁺ concentration distribution at 200 μm above anode.

The Cu²⁺ concentration distribution uniformity was evaluated by calculating the Cu²⁺ concentration distribution standard deviation at the y position of 200 μm, and the calculation results are shown in Fig. 3(b) under the three experimental conditions. Since these values under the three experimental conditions were almost the same at 0 s, the light reflection on the uneven anode surface did not affect the concentration uniformity evaluation results.

The standard deviation monotonically increased with time under the DC condition and the DC+MF condition. This is because the higher Cu²⁺ concentration near the side parts of the anode surface reduced the Cu²⁺ concentration distribution uniformity with time. In contrast, the standard deviation under the DC+AC+MF condition slight decreased at 20 s, because the relatively dark solution by dissolving Cu²⁺ ion near the anode surface suppressed the light reflection caused by the uneven anode surface. Its value was almost constant from 40 s to 60 s, because of the suppression of higher Cu²⁺ concentration near the side parts of the anode surface. In addition, its magnitude was the smallest among the three conditions. This means that the most uniform Cu²⁺ concentration distribution was performed under the DC+AC+MF condition.

3.2. Velocity Measurement Results

Figure 4 shows the electrical current flow curves and the electromagnetic force directions under the DC+MF and the DC+AC+MF conditions. Near the middle part of the anode surface, the current flowed perpendicular to the anode surface. However, near the left and right parts, the current directions were not perpendicular to the anode surface because of the length difference between the anode and the cathode,²⁶⁾ as indicated by the dashed curves. Therefore, the directions of the excited electromagnetic forces near the left, middle and right parts of the anode surface were oblique upward, parallel, and oblique downward to the anode surface respectively, as marked by the dashed arrows.

Fig. 4.

Schematic of electrical current flow curves and electromagnetic force directions from 0 s to 60 s.

Figure 5 shows the liquid flow pattern under the DC+MF condition. A circulating macro-scale flow was excited in the whole vessel within a few seconds, and its direction was counterclockwise from the front view. This is because of the 0.02 T magnetic flux density difference between near the anode and near the cathode as has been shown in Fig. 1. Therefore, the liquid with initial 0.3 mol/L Cu²⁺ concentration flowed from the bulk region into the vicinity of the anode surface, and the average Cu²⁺ concentration decreased compared to that under the DC condition as shown in Fig. 3(a).

Fig. 5.

Liquid flow pattern under DC+MF condition.

An asymmetric circulating macro-scale flow pattern was observed in the whole vessel as shown in Fig. 5, due to the asymmetric electromagnetic force direction distribution as shown in Fig. 4. Near the middle part of the anode surface, flow direction was parallel to the anode surface. This contributes to uniform the Cu²⁺ concentration distribution. Therefore, a relative uniform Cu²⁺ concentration distribution near the middle part was observed, as shown in Fig. 2(b). On the other hand, stagnant zones formed near the side parts of the anode surface. Noteworthy, the stagnant zone near the left part was larger than that near the right part. And the enlargement of the stagnant zone near the left part was observed. One of the reasons for the formation of the stagnant zones might be the 180 μm height difference between the insulator and the anode.³¹⁾ An additional reason for that near the left part of the anode surface is considered as the oblique upward electromagnetic force as shown in Fig. 4. And near the right part, the upward component of the circulating macro-scale flow according to the mass conservation law is supposed to be another reason for the formation of the stagnant zone. Because of the formation of the stagnant zones and the current concentration near the side parts, higher Cu²⁺ concentration was observed near the side parts as shown in Fig. 2(b). Since the electrical conductivity of the aqueous solution has positive relation with the Cu²⁺ concentration, the current concentration near the side parts was enhanced. This further enhanced the dissolution of the Cu²⁺ concentration from the left part of the anode. Furthermore, the electromagnetic force near the side parts increased. This might result in the enlargement of the stagnant zone near the left part, which also enhanced the increase of Cu²⁺ concentration near the left part. Therefore, Cu²⁺ concentration obviously increased with time near the left part compared to that near the middle and right parts, as shown in Fig. 2(b).

Figure 6 shows the maximum and minimum velocities at the y position of 200 μm under the DC+MF condition, which were evaluated by measuring the maximum and the minimum velocities of the tracer particles. Large velocity difference between the maximum and minimum velocity was observed. Because the liquid velocity at the side walls of the vessel is 0 μm/s, the velocity of the particles moving in the center of the vessel was larger than that moving close to the front or back walls of the vessel. This means that the maximum and the minimum velocities represent the characteristic velocities of the circulating macro-scale flow in the center and close to the side walls of the vessel respectively. Thus, the minimum velocity was close to 0 μm/s. On the other hand, the maximum velocity increased with time due to the development of the circulating macro-scale flow. Because of the limitation of flow close to the side walls, a high Cu²⁺ concentration increase rate close to the side walls of the vessel compared to that in the center part in the z-direction is expected. As the Cu²⁺ concentration distribution evaluation results should be the average Cu²⁺ concentration in z-direction, the higher Cu²⁺ concentration increase rate close to the side walls of the vessel is supposed as a reason for the increase of the Cu²⁺ concentration under this condition as shown in Fig. 2(b).

Fig. 6.

Maximum and minimum velocities at y position of 200 μm under DC+MF condition.

Figure 7 shows the serial change of the liquid flow pattern with time under the DC+AC+MF condition. Similar to that under the DC+MF condition, the counterclockwise asymmetric circulating macro-scale flow was excited in the whole vessel within a few seconds, because of the same average electromagnetic force distributions between the DC+MF and the DC+AC+MF conditions. Furthermore, particle motions in the positive and negative x-directions were simultaneously observed from about 5 s between just above the insulator surface and the left and right parts of the anode surface. This indicates that circulating micro-scale flows were excited from about 5 s as shown in Fig. 7(a). These flows were expressed as “anode-insulator circulating micro-scale flows” in this research. Besides, the liquid flow pattern changed with time. The liquid flow pattern around 50–60 s is shown in Fig. 7(b). The anode-insulator circulating micro-scale flow near the right part of the vessel shrank and was only observed above the right side insulator surface. However, no obvious shrinkage of the anode-insulator circulating micro-scale flow was observed near the left part. In addition, another circulating micro-scale flow was excited just above the right part of the anode surface at around 50 s. This flow was expressed as “local circulating micro-scale flow” in this research. The excitation of the circulating micro-scale flows indicates that the liquid flow pattern changed by imposing the AC current. However, the mechanism for the excitation of the circulating micro-scale flows near the side parts of the anode surface and the time evolution of the local circulating micro-scale flow were not confirmed in the 60 s experimental time, which will be clarified in the future work.

Fig. 7.

Liquid flow pattern under DC+AC+MF condition around (a) 5–50 s and (b) 50–60 s.

The circulating macro-scale flow contributes to uniform the Cu²⁺ concentration distribution especially near the middle part of the anode surface. On the other hand, because of the excitation of the anode-insulator circulating micro-scale flows near the side parts of the anode surface, the liquid with initial 0.3 mol/L Cu²⁺ concentration flowed from just above the left and right part of the insulators to the left and right parts of the anode surface. This suppressed the increase of the Cu²⁺ concentration near the side parts and uniformed the Cu²⁺ concentration distribution. Besides, the local circulating micro-scale flow also contributes to uniform the Cu²⁺ concentration distribution by mixing the liquid with higher Cu²⁺ concentration with the liquid with lower Cu²⁺ concentration. Owing to the excitation of the circulating micro-scale flows, the higher Cu²⁺ concentration near the side parts was not observed, and the Cu²⁺ concentration distributed more uniformly compared to that under the DC+MF condition, as shown in Fig. 2(c).

Figure 8 shows the maximum and minimum velocities at the y position of 200 μm under the DC+AC+MF condition. Due to the observation of the negative x-direction particle motion, the minimum velocity was the negative value. This means that the minimum velocity represents the characteristic velocity of the circulating micro-scale flow (the anode-cathode circulating micro-scale flow or the local circulating micro-scale flow). The absolute value of the minimum velocity increased from 20 s to 40 s and slightly decreased from 40 s to 60 s, while the reason has not been clarified. On the other hand, because the maximum velocity shown in Fig. 8 was much larger than the absolute value of the minimum velocity, and the absolute values of the positive and negative velocity components of the circulating micro-scale flow should be in the same magnitude, the maximum velocity represents the characteristic velocity of the circulating macro-scale flow. The maximum velocity increased with time, owing to the development of the circulating macro-scale flow. Its magnitudes at 20 s was about 1000 μm/s, which was much larger than that of about 400 μm/s under the DC+MF condition as shown in Fig. 6. In addition, the maximum velocity difference between 20 s and 60 s was about 100 μm/s, which is smaller than that of about 200 μm/s under the DC+MF condition. These mean that the development of the circulating macro-scale flow was enhanced by imposing the AC current. The enhancement of the circulating macro-scale flow development might be another reason for the suppression of the Cu²⁺ concentration increase compared to that under the DC+MF condition as shown in Fig. 2(c).

Fig. 8.

Maximum and minimum velocities at y position of 200 μm under DC+AC+MF condition.

4. Conclusion

In this research, the Cu²⁺ concentration distribution near the whole vicinity of the anode surface under the force imposition with or without an oscillation component, and the mechanism for the Cu²⁺ concentration distribution change were studied. The following results can be achieved:

(1) By imposing the force without oscillating component, the increase of the Cu²⁺ concentration was suppressed compared to that without force imposition. The average Cu²⁺ concentration increased with time and the Cu²⁺ concentration distribution uniformity decreased with time.

(2) By imposing the force with oscillating component, the increase of the Cu²⁺ concentration was further suppressed, and the average concentration was almost constant with time evolution. The Cu²⁺ concentration distribution was the most uniform.

(3) The development of the circulating macro-scale flow was enhanced by imposing the oscillating force. This contributes to suppress the increase of the Cu²⁺ concentration.

(4) The liquid flow pattern changed by imposing time-varying force. That is, circulating micro-scale flows were excited near the side parts of the anode surface. These flows contribute to suppress the increase of the Cu²⁺ concentration and uniform the Cu²⁺ concentration distribution.

Acknowledgments

The first author gratefully acknowledges the financial support from China Scholarship Council (No. 201806060147), which has sponsored his study at the Hokkaido University in Japan.

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