The primary current distribution and the resistance of a modified Hull cell are calculated by using conformal mapping technique coupled with numerical evaluation of the resulting integral equations. An approximate analytical expression for the primary current distribution of a modified Hull cell is presented. The primary current distribution along the cathode surface is noticed varying in controlled manner as a function of position on the substrate. The current distributions (primary, secondary, and tertiary) in the cell have also been calculated at different applied average current densities (2, 4.1, and 8.2 mA cm^{−2}) through numerical simulation by using finite element based software. The numerical simulation result of the primary current distribution is then compared with the analytical solution and a good match is found. Experimentally, single Cu metal electrodeposition is carried out at different applied average current densities (2, 4.1, and 8.2 mA cm^{−2}) in a modified Hull. The current distribution (primary, secondary, and tertiary) results obtained from the numerical simulation are compared with the experimental results and a satisfactory match is found. Surface morphology of the Cu deposits is examined using scanning electron microscopy (SEM).

Electrodeposition, a versatile, cost effective, and simple technique, is used to fabricate metallic coatings. The electrodeposition process parameters especially current density can affect the surface morphologies, chemical compositions (in case of alloys), and properties of the coatings which is clearly described in many reports of the literature [

In electroplating, the given electrochemical cell configuration is first needed to be understood with calculation of the current density distribution in the cell by taking into account other effects such as electrochemical reaction kinetics and mass transfer. However, to analyze and understand the electrochemical system, the initial step is the calculation of primary current distribution (PCD) along the electrode surface and primary resistance of the electrochemical cell in which the surface overpotential is neglected and the equipotential surface of the solution adjacent to the electrode is assumed [

The primary current along the electrode and the potential distribution in the electrolyte are calculated from solution of Laplace’s equation (

In the literature, many articles [

Conformal mapping technique has long been used to calculate current distributions in different electrochemical cell geometries [

Generally, COMSOL Multiphysics, a finite element based software, is used to study the current and the potential distribution along the cathode in several electrochemical cells [

Generally used pure Cu electrodeposition from acid sulfate electrolytes is considered to compare with the simulated results [

In the present work, the main objective is the calculation of the primary current distribution and primary resistance of the cell of the modified Hull cell using conformal mapping technique. The development of the analytical expression for the PCD and the calculation of primary cell resistance value using conformal mapping technique (Schwartz-Christoffel Transformations) are presented. Further, numerically current density distributions (PCD, secondary (SCD), and tertiary (TCD)) in the cell during Cu electrodeposition carried out at different applied average current densities are investigated and its PCD is compared with the analytical PCD curve. Finally, Cu is deposited experimentally through pulsed electrodeposition in a modified Hull cell at similar applied average current densities and its normalized thickness distributions is compared with the numerically simulated current distribution (PCD, SCD, and TCD) results.

Modified Hull cell, a simple small electrodeposition cell, has a trapezoidal structure which consists of cathode (dimensions: 7 × 3.5 cm^{2}) placed at an angle of 51.5° with anode (dimensions: 4.35 × 3.5 cm^{2}) and two insulating walls (schematic diagram is shown in Figure

Schematic diagram of the modified Hull cell geometry.

In contrast to standard Hull cell geometry, a modified Hull cell with an angle between the electrodes of 51.5° and modified dimensions of anode and cathode is designed to fabricate the Cu films in the present study. As stated in the introduction, the study of current distribution in a designed modified Hull cell is of prime concern and is necessary to be known before electrodeposition of metals.

The dimensions of the modified Hull cell before and after scaling are shown in Figures

The schematic representation of the modified Hull cell geometry in the three coordinate systems used in the conformal mapping: (a)

The conformal mapping technique is a powerful tool and is used to obtain analytical solution of PCD of a modified Hull cell. The actual cell geometry, the trapezoid (

when

when

Though Cu films are fabricated experimentally through pulsed electrodeposition in a modified Hull cell, the average current densities used in the experimental PED are only used in the simulation of current distributions during Cu electrodeposition.

Figure

Geometry of the modified Hull cell.

In the absence of the concentration gradients in the electrolyte, Ohm’s law offers the relationship between local current density (

The potential distribution in the electrolyte is calculated from the solution of Laplace’s equation:

By neglecting the concentration effects within the diffusion layer, the current distribution resulted from both effects such as the reaction kinetics on electrode and the geometry of the cell called secondary current distribution (SCD). The formation of different electroactive species with Cu^{2+} ions from citrate baths depending on the plating bath parameters such as solution pH and total citrate concentration is described in detail elsewhere [^{2+} ions are considered because numerical solution is not achieved with the finite element based software when Cu citrate complex (Cucit^{−}) singly and both the combination of Cu^{2+} and Cu citrate complex (Cucit^{−}) are taken into consideration. Hence, the following Cu deposition reactions (

The following equation is used to calculate the overpotential (

Equilibrium potential for Cu electrode reaction

When geometry effect, electrode kinetics effect, and concentration effects within the diffusion layer are considered, the resulted current distribution is called tertiary. Nernst diffusion layer model is used and is assumed to be constant thickness of 30 ^{2+}, is small and its effect is also neglected. Therefore, the mass transport mechanism of electroactive species within the diffusion layer is described by diffusion only,^{2+}), and ^{2+}).

The concentration distribution within the diffusion layer is calculated by using Laplace’s equation:

Meshing of the modified Hull cell.

For numerical simulation of Cu electrodeposition using COMSOL, the Electrochemistry Module (primary (secondary) current distribution (siec) interface) is used to simulate PCD, SCD, and TCD. After calculating the PCD, current distribution is changed from primary to secondary in the Current Distribution Type section to solve the SCD. The combination of secondary current distribution (given as “siec” in COMSOL) and Transport of Diluted Species (given as “chds” in COMSOL) physics is used to calculate TCD and concentration distribution in the diffusion layer.

For Cu electrodeposition, stainless steel of 7 × 3.5 cm^{2} and pure Cu (99.9%) of 4.35 × 3.5 cm^{2} are used as the substrate and the anode materials, respectively. The substrate materials are polished mechanically to generate mirror like surface followed by ultrasonic cleaning in acetone for 5 min and rinsing with deionized water. The angle, closest distance, and farthest distance of the oblique angled cathode to vertically positioned anode in the modified Hull cell are 51.5° (approximately), 1 cm and 6.5 cm, respectively. Pulsed electrodeposition (PED) is used to fabricate pure Cu films onto stainless steel substrate in a modified Hull cell of 1 liter capacity plating bath composed of 0.02 M CuSO_{4}·5H_{2}O and 0.2 M Na_{3}C_{6}H_{5}O_{7}·2H_{2}O. The bath is operated at 55°C and the operating pH is 4. The agitation of the bath is maintained at 200 rpm through magnetic stirrer. The pulsed electrodeposition experiments are performed at on-time of 1 ms, off-time of 10 ms, and applied average current densities of 2 (applied current of 0.05 A), 4.1 (applied current of 0.1 A), and 8.2 mA cm^{−2} (applied current of 0.2 A) for 60, 30, and 15 min, respectively.

The surface morphologies of the fabricated Cu films are investigated on Zeiss Supra field emission scanning electron microscope through secondary electron imaging operating at 10 kV of accelerating voltage.

Thickness of the deposited films is measured with optical profiler (Zeta Optical Profiler-20). Thickness measurements are taken at a regular distance of 1 cm from LCD end to HCD end of the substrates. Thickness data at each position on the working electrode is the average of the four measurement points taken horizontally on both sides from the centre of the substrates.

The parameters (considered for the simulation which is listed in Table

Transport and kinetic parameters used in the numerical simulation.

Parameter | Value | Reference |
---|---|---|

Electrolyte conductivity (S cm^{−1}) |
0.5 | Measured |

Applied average current densities (^{−2}) |
−2, −4.1, and −8.2 | |

Exchange current density (^{−2}) |
3.92 × 10^{−7} |
[ |

Tafel slope ( |
−0.13 | [ |

Diffusion coefficient (cm^{2} s^{−1}) |
1 × 10^{−5} |
[ |

The conductivity of the electrolyte (^{−1}).

The asymptotic or approximate solution for the analytical expression of the PCD in the trapezoidal geometry type electrochemical cells is presented in [

The parameters of the modified Hull cell are given in the following.

The form of an estimation of current distribution in the modified Hull cell by substituting the calculated constants (

Comparison of current density distribution in a modified Hull cell [solid red color line joining solid stars] with an empirical formula [black color line] [

The PCDs obtained from (

The dimensionless ohmic resistance (

For modified Hull cell, the calculated dimensionless ohmic resistance is found equal to 0.5223.

The current density distribution lines are drawn in the cell to confirm high and low values of current densities at HCD end and LCD end of the working electrode, as shown in Figure

The distribution of electrolyte current density stream lines in modified Hull cell.

Figure ^{−2} as a function of position on the working electrode. It is increasing from LCD end to HCD end of the working electrode as position on the working electrode increases but it is found constant for all applied average current densities: 2, 4.1, and 8.2 mA cm^{−2}. It confirms that PCD provides nonuniform distribution which covers wide range of current densities from LCD end to HCD end of the working electrode. Therefore, it gives a chance to study the influence of wide range of current densities on the characteristics of the deposit in a single experiment. From these results, the PCD dependence on the geometry of the cell itself is clearly known.

Primary current distribution for modified Hull cell during Cu electrodeposition carried out at different applied average current densities of 2, 4.1, and 8.2 mA cm^{−2}.

Inset figure in Figure

Figure

Comparison of analytical and numerical solutions for primary current distribution for modified Hull cell.

By adding the effect of electrode reaction kinetic resistance to PCD in the numerical simulation, the resulted secondary current distributions obtained for different applied average current densities, 2, 4.1, and 8.2 mA cm^{−2}, are shown in Figure ^{−2} than that of 8.2 mA cm^{−2}. The nonuniformity of current distribution increases with increasing applied average current density in the case of SCD. Therefore, SCD becomes PCD at high applied average current density where the electrode reaction kinetics is very fast (approximately equal to infinity).

Secondary current distribution for modified Hull cell during Cu electrodeposition carried out at different applied average current densities of 2, 4.1, and 8.2 mA cm^{−2}.

A single dimensionless parameter, the Wagner number (

The equation related to the Wagner number evaluated at the Tafel limit is given by

It is a measure of the uniformity of current distribution on the cathode. The most uniform current distribution resulted when the Wagner number reaches its highest value [^{−2}) to 4.6 (2 mA cm^{−2}).

By the addition of concentration effects in the diffusion layer to other effects such as geometry and electrode reaction kinetic resistance in the numerical simulation, the resulted tertiary current distributions obtained for different applied average current densities, 2, 4.1, and 8.2 mA cm^{−2}, are shown in Figure ^{−2} because of the variation in the concentration of copper ions within the diffusion layer. These numerically calculated PCD, SCD, and TCD are then compared with the experimental data.

Tertiary current distribution for modified Hull cell during Cu electrodeposition carried out at different applied average current densities of 2, 4.1, and 8.2 mA cm^{−2}.

To compare the simulated curves with the experimental results, Cu deposition from the citrate based plating bath is carried out at different average current densities: 2, 4.1, and 8.2 mA cm^{−2} for 60, 30, and 15 min, respectively. The thickness of the deposit (

Comparison of experimental determined normalized deposit thickness distribution (^{−2}.

For the Cu film fabricated at 2 mA cm^{−2} (see Figure ^{−2} (see Figure

Plot of average current efficiency versus applied average current density.

In case of Cu film fabricated at 4.1 mA cm^{−2} (see Figure

For the Cu film fabricated at 8.2 mA cm^{−2} (see Figure

The Cu films are fabricated through pulsed electrodeposition in a modified Hull cell for 60, 30, and 15 min at different applied average current densities 2, 4.1, and 8.2 mA cm^{−2}. It is observed that, at all applied average current densities of 2, 4.1, and 8.2 mA cm^{−2}, the deposit is bright throughout the coating surface over current density range of 1.1–5.2, 0.2–10.7, and 0.4–21.4 mA cm^{−2} (current density at the both edges of the cathode is not considered). At an applied average current density of 2 mA cm^{−2}, the film is not formed completely at the center of the substrate in the current density range 0.1–1.1 mA cm^{−2}. The bubbles are observed at HCD end of the substrate in the Cu films fabricated at applied average current densities of 4.1 and 8.2 mA cm^{−2}. The reddish color of the deposit surface is changed from light to dark in 2 to 8.2 mA cm^{−2}.

The surface morphologies of Cu electrodeposits fabricated by pulsed electrodeposition from citrate-sulfate electrolyte at applied average current densities ≈2, 4.1, and 8.2 mA cm^{−2} are investigated at different positions of the working electrode using SEM and are shown in Figures

SEM micrographs taken at different regions on the substrate: 1 cm, 3 cm, and 6 cm showing the surface morphology of the electrodeposited Cu film fabricated at 2 mA cm^{−2} (a)–(c); 4.1 mA cm^{−2} (d)–(f); and 8.2 mA cm^{−2} (g)–(i).

The surface morphology of electrodeposited Cu in different applied average current densities (2 (a)–(c), 4.1 (d)–(f), and 8.2 mA cm^{−2} (g)–(i)) is found varied from smooth to granular to globular with rough surface (because of hydrogen gas evolution), indicating mass transfer electrodeposition mechanism. It is expected that hydrogen gas evolution reaction consumes some portion of the applied currents which should result into low current efficiencies for Cu electrodeposition carried out at applied average current densities of 4.1 and 8.2 mA cm^{−2}.

Analytical calculation of primary current distribution and primary cell resistance of the modified electrochemical Hull cell is performed using conformal mapping technique. The analytical solution of PCD of the modified Hull cell is then compared with the standard Hull cell and the empirical formula and a good agreement is found.

Numerical simulation of current distributions (primary, secondary, and tertiary) during the electrodeposition of Cu is carried out at different applied current densities: 2, 4.1, and 8.2 mA cm^{−2} in a modified Hull cell using COMSOL Multiphysics software.

For comparison, Cu film with thickness gradient is fabricated through pulsed electrodeposition in a modified Hull in a single experiment from the citrate based electrolyte.

For Cu electrodeposition, the experimentally measured normalized thickness distribution (^{−2}. At 2 mA cm^{−2}, higher experimental thickness distribution of Cu deposit is observed which might be due to organic inclusions increasing its weight, whereas, in case of the Cu film fabricated at 4.1 mA cm^{−2}, the experimental thickness distribution almost corresponds similar to the simulated SCD and TCD curves. For the Cu film fabricated at 8.2 mA cm^{−2}, the experimental thickness distribution goes above and below the simulated SCD and TCD curves when compared from LCD and HCD end of the electrode. Overall, in all the cases, experimental dimensionless deposit thickness distribution followed the simulated SCD and TCD curves in a reasonable way.

Concentration at the surface of the working electrode (mol cm^{−3})

Diffusion coefficient of electroactive species (Cu^{2+}) (cm^{2} s^{−1})

Faraday constant (C mol^{−1})

Distance between the electrodes at HCD end (see Figure

Applied average current density (A cm^{−2})

Partial current density obtained from the electrode reaction, ^{−2})

Local current density at any point in the modified Hull cell (A cm^{−2})

Normalized current distribution

Exchange current density at bulk concentrations for reaction, ^{−2})

Electrode reaction and imaginary number,

Quantities used in the conformal mappings (see (

Length of the cathode (cm)

Molar mass of Cu (g mol^{−1})

Flux of electroactive species (Cu^{2+}) (mol cm^{−2} s^{−1})

Number of electrons transferred in an electrode reaction, ^{−2})

Universal gas constant (8.3143 J mol^{−1}K^{−1})

Dimensionless cell resistance (see Figure

Total deposition time (s)

Absolute temperature (K)

Equilibrium potential for the reaction,

Wagner number for a Tafel kinetic approximation

(Dimensionless) real and imaginary components of

Complex coordinate systems used in the conformal mappings (see Figure

High current density

Low current density

Primary current distribution

Pulsed electrodeposition

Secondary current distribution

Tertiary current distribution.

Cathodic transfer coefficient of reaction,

Cathodic Tafel slope (V)

Overpotential for the reaction,

Angle formed at a corner of the working electrode and insulating wall (radians) (see Figure

Conductivity of the electrolyte (S cm^{−1})

Experimental thickness of the electrodeposited Cu thin film (

Theoretical thickness of the electrodeposited Cu thin film (

Normal to the cathode surface (cm)

3.1415926…

Density of Cu (8.93 g cm^{−3})

Potential (V)

Electrolyte potential (V)

Electrode potential (V).

Bulk

Surface.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are indebted to Dr. Challa Subrahmanya Sastry (Department of Mathematics, IITH) and Dr. Jampana Phanindra (Department of Chemical Engineering, IITH) for the technical discussions to calculate primary current distribution analytically using conformal mapping technique. The authors also would like to thank BVRSN Prasad, K. Manjunath, and M. Srinivas for their help in developing the MATLAB program. This research work has been financially supported by Council of Scientific and Industrial Research (CSIR) (CSIR Sanction no. 22(0585)/12/EMR-II), India.