The Establishment of Current Transient of Nucleation and Growth under Diffusion-Controlled Electrocrystallisation: A Microreview

Abstract

In this microreview, the importance of electrochemical phase formation is outlined. Further, the establishment of current transition during electrochemical nucleation and growth is explored. First, the microreaction kinetic process of electrocrystallisation and the rate-controlling steps of electrodeposition are carefully discussed. Then, the current transient formulation under electrochemical polarisation-controlled electrocrystallisation is summarised. Finally, a summary of the process for calculating the current transient formulation of nucleation and growth under diffusion-controlled electrocrystallisation is explained in detail.

1. Introduction

Electrocrystallisation refers to the process in which metal cations are reduced by electric discharge under the action of an electric field, resulting in crystal nucleation of metal atoms and metal formation [1,2,3]. The crystallisation process of electrocrystallisation nucleation, which directly determines the composition, structure and properties of sediments, is the core problem in electrochemical research and the cornerstone of electrodeposition technology development [4,5]. Localised electrochemical deposition (LECD), also known as microanode-guided deposition technology, is a microadditive manufacturing technology that facilitates controllable deposit growth on cathode surfaces and control of ion reduction crystallisation growth in electrolytes [6]. To date, metal microstructures with high aspect ratios and complex geometric shapes have been prepared by LECD, such as helix (for special resonators) and cantilever (for electronic circuit interconnection) [7]. At the same time, LECD can deposit metals, alloys, conductive polymers and other materials with high deposition rates and fine crystallisation. LECD has demonstrated strong potential as a method for small-scale electroplating interconnection technologies [8,9,10].

The core challenge in LECD and other electrochemical fine manufacturing technologies is accurately controlling the electrodeposition crystallisation growth process [11,12,13,14]. First, the crystal growth pattern in electrodeposition should be studied. The development of LECD and other micro-manufacturing technologies has necessitated systematic and in-depth research into the electrochemical phase formation of materials. One of the most important topics in the study of electrochemical phase formation is the study of electrochemical phase formation using the change of the i–t curve (chronoamperometry) during electrocrystallisation.

Theoretical and experimental studies of the mechanism of electrocrystallisation have been conducted in the past 100 years [15,16]. As early as 1903, Cottrell [17] proposed an equation (Cottrell equation) describing the change in current relative to time in controlled potential experiments. The diffusion pattern was used to study the relationship between ion diffusion on the surfaces of flat and spherical electrodes under constant potential. In 1928, Stranski et al. [18] studied energy differences in the electrocrystallisation of new phases at the solid-liquid interface at different sites, indicating that new phases are more likely to be generated at the step or inflection point of basement microscopic surfaces. Subsequently, Conway et al. [19] proposed that the preferential growth of a step or corner lattice is inevitably accompanied by surface diffusion of adsorbed atoms. In 1958, Fleischmann M. et al. [20,21,22] introduced spiral dislocation crystal growth theory into electrocrystallisation theory; by obtaining the timing current curve of PbO₂ deposition, they extracted information about the nucleation and growth process of deposition, so as to explain the spiral dislocation growth observed on the surface of electrodeposited copper. In 1983, Hills and Scharifker et al. [23,24,25] proposed a theoretical model of three-dimensional hemispherical infinite diffusion mass transfer electrodeposition crystallisation based on the study of microscopic dynamics of electrodeposition crystallisation and established the current time function of the electrodeposition crystallisation process corresponding to the experimental i–t curve. All electrochemical researchers focused on the study of the current transient and the microscopic process of electrocrystallisation, which provides a bridge of in situ electrical calculation for the study of the microscopic process of electrodeposition crystal growth by using the experimental i–t curve [26,27]. At present, the study on the microscopic process of electrodeposition crystallisation growth by the experimental i–t curve is still advancing, but there are many differences between the theoretical description and the actual process.

This paper aims to systematically summarise the main theoretical ideas of electrochemical phase formation and the establishment of current transition during electrochemical nucleation and growth. This paper provides systematic basic theory and research methods for studying the microdynamic process of electrodeposition, electrodeposition regulation theory and emerging fine electrochemical manufacturing technologies, such as LECD.

2. Electrochemical Phase-Forming Process

Electrocrystallisation refers to the process in which metal cation discharge is reduced under the action of an electric field and results in the nucleation, growth and crystallisation of metals. During the electrodeposition of metal ions, the state of a cathode surface is constantly updated because newly generated metal is constantly attached to the cathode surface. The electrodeposition of metal atoms is affected by an electric field on a cathode surface. Owing to the interaction between these processes, the kinetic process of metal ion reduction on a cathode surface is extremely complicated [28,29,30]. The deposition of metal ions on a cathode surface is a multistep chain reaction process. Its microscopic process is shown in Figure 1 [27,31].

Figure 1a shows a series of steps in the electrochemical crystallisation of metal ions on an electrolyte–electrode surface: ion migration to the electrode surface, ligand removal of complex ions on the electrode surface, adsorption atom generation by ion acquisition through electron reduction and atom migration on the electric surface or directly to the crystallisation site for crystal growth [32,33]. The migration speed of metal ions in electrolytes, the overpotential of an electrode and the mobility of adsorbed atoms or atomic groups on an electrode surface affect the microdeposition and growth state of metals [34]. Figure 1b–e illustrate the influence of overpotential η and relative surface mobility factor α on the size distribution, quantity density and surface coverage of deposited metal clusters during crystallisation. The rates of cluster formation and growth increase with η. Moreover, the mobility of large clusters and particle size dispersion decreases with increasing relative surface mobility factor, and the number of clusters increases with α [31].

Electrocrystallisation is a multistep chain process, and each link and step affect the whole chain reaction. To study the rate of whole chain reaction, researchers usually examine the step with the highest reaction resistance and the slowest reaction speed as the speed control step in electrodeposition [27]. Rate-controlling steps reflect the entire electrocrystallisation process. Theoretically, any step of electrocrystallisation can become the rate-controlling step [35]. In the actual electrocrystallisation process, nucleation and growth can be broadly classified into two categories [36]: electrochemical polarisation-controlled, in which the nucleus growth rate is limited by the rate at which ions can be incorporated into a new phase, and diffusion-controlled, in which nucleus growth is limited by the rate at which a material is transported through the electrolyte and to an electrode surface [37,38]. Next, the electrochemical polarisation-controlled and diffusion-controlled electrocrystallisation processes will be discussed in the following sections.

3. Current Transient of Electrochemical Polarisation-Controlled Electrocrystallisation

In electrochemical reactions controlled by an electrochemical polarisation step, the reaction rate is controlled by overpotential. The electrochemical reaction under the control of the electrochemical polarisation step is subject to the law of the electrochemical thermodynamic reaction rate [39]. The electrochemical crystallisation reaction rate and reaction current are subject to the Arrhenius formula and Butler–Volmer equation, respectively.

The Arrhenius formula [40] proposed by Svante August Arrhenius in 1889 describes the relationship between reaction velocity and activation energy, as shown in Equation (1).

$$\sigma ={\sigma }_0\cdot exp-\frac{H}{RT}$$

(1)

where $\sigma$ is the rate constant, ${\sigma }_0$ is the pre-exponential (“frequency”) factor, $H$ is the activation energy of the reacted particles, $R$ is the gas constant, and $T$ is the temperature [41].

The influence of ion migration and mass transfer is not considered in the calculation of the electrocrystallisation reaction controlled by electrochemical polarisation. Combined with the Arrhenius formula, the Bulter–Volmer equation can describe the overall reaction rate, reaction charge and current on an electrode surface by summing the positive and negative reaction rates [42], as shown in Equation (2).

$$J=J_0\left(e^{\frac{{\alpha }_a\eta zF}{RT}}-e^{-\frac{{\alpha }_c\eta zF}{RT}}\right)$$

(2)

where $J$ is the current density, $J_0$ is the exchange current density, ${\alpha }_a, {\alpha }_c$ are the transfer coefficient, $\eta$ is the overpotential, $R$ is the gas constant, $T$ is the temperature, z is the number of charges of substances during the electrodeposition reaction process, F is the Faraday constant and $zF$ represents the molar charge of the electrodepositing species, respectively [43].

The Butler–Volmer formula shows that the difference between positive and negative reaction rates on an electrode surface depends on exchange current density and polarisation overpotential under the control of electrochemical polarisation [44]. When the overpotential is large, the difference between the two terms in the Butler–Volmer formula increases accordingly. A small term is negligible, and Equation (2) is in agreement with the quantitative overpotential–current density relationship described by Tafel’s empirical formula proposed in 1905 [45]. In addition, according to Equation (2), a change in voltage can exponentially accelerate the change in current. When the voltage is extremely large, the current approaches infinity (as shown in Figure 2c), and this effect is obviously impossible.

The existing study showed that ions deposited on an electrode surface are consumed rapidly at a high overpotential, and the reaction rate of electrodeposition crystal growth is determined by the diffusion supply rate of reactive ions [36]. As mentioned above, in the actual electrocrystallisation process, nucleation and growth can be broadly classified into two categories: electrochemical polarisation-controlled and diffusion-controlled. The current transition function under diffusion-controlled discharge deposition will be discussed in the following sections.

4. Current Transient of Diffusion-Controlled Electrocrystallisation

In electrocrystallisation controlled by diffusion, concentration polarisation (electrode polarisation phenomenon that occurs when the controlling step is the mass transfer step in the electrolyte) replaces electrochemical polarisation and becomes the control step of an electrochemical reaction, that is, the speed of the diffusion of reactive ions in electrolytes to electrode surfaces becomes the rate-controlling step of electrodeposition reactions. Given that the diffusion rate and flux of ions in electrolytes change with time, diffusion current usually changes with time.

In 1903, Cottrell [17] used the diffusion law to study the transition pattern of electric flux on the surface of an electrode subjected to the unsteady diffusion of a planar electrode; Equation (3) was used to represent change in the diffusion current of the planar electrode.

$${\mathrm{i}}_{L,d}=nFAD{\left(\frac{\mathsf{\partial }c}{\mathsf{\partial }x}\right)}_{x=0}$$

(3)

where ${\mathrm{i}}_{L,d}$ is the limiting diffusion current, $zF$ represents the molar charge of the electrodepositing species, $A$ is the electrode area, $D$ is the Diffusion coefficient, $c$ is the ion concentration, and $x$ represents the distance from the electrode surface, respectively.

Identifying the process involved in the change in electrode surface concentration over time is necessary to determine electrode surface current. According to Fick’s second law [50], as shown in Equation (4).

$$\frac{\mathsf{\partial }C}{\mathsf{\partial }t}=D\frac{{\mathsf{\partial }}^{2}C}{\mathsf{\partial }{x}^2}$$

(4)

where $C$ is the bulk concentration, and $t$ is the time for electrodepositing. To solve Equation (4), the following boundary conditions were adopted:

When there is no electrochemical reaction on the electrode surface, $t=0;\to C_s=C_{\infty }$. When $t\ge 0;\to \underset{x\to \propto }{\lim }C=C_{\infty }$. And $t>0, x=0;\to C_x=0$; in other words, the ions that participate in the reaction are consumed completely on the electrode surface.

By applying Fick’s second law to the Laplace transform of time t, the change rule of concentration can be obtained, as shown in Equation (5).

$$c=c_{\infty }erf\left[\frac{x}{2{\left(Dt\right)}^{1/2}}\right]$$

(5)

where the function ($\mathit{erf}\left[ \right]$) represents the error function. Through a series of mathematical calculations, the variation law of the concentration gradient on the electrode surface over time can be obtained at x = 0, as shown in Equation (6).

$${\left(\frac{\mathsf{\partial }c}{\mathsf{\partial }x}\right)}_{x=0}=\frac{c_{\infty }}{{\left(\pi Dt\right)}^{1/2}}$$

(6)

where $c_{\infty }$ is the bulk concentration. Equation (6) is substituted for Equation (3). After the potential step occurs on the electrode surface, the expression of the unsteady diffusion current on the electrode surface over time can be obtained through Equation (7), also called the Cottrell equation [51].

$${\mathrm{i}}_{L,d}=nFA{c}_0{\left(D/\pi t\right)}^{1/2}$$

(7)

The Cottrell equation describes the relationship between ion consumption on the surface of an electrode and diffusion ion flux formation current over time after a step potential is applied on the surface of a planar electrode subjected to diffusion-controlled electrocrystallisation [17]. Variation in the current obtained by Equation (7) over time represents the current response law of an electrode surface completely supplied by the migration of diffusion-controlled ions, that is, the current response of the pure diffusion-controlled reaction at the potentiostatic method. The equation does not consider the details of the electrode surface, such as the distribution and density of the electrocrystalline growth nuclei, and only focuses on the change of the total diffused energy on the surface of the planar electrode with time [52,53].

Later, the nucleation and growth process of electrocrystallisation were investigated on the electrode surface. More detailed studies and calculations of the current function were carried out [49,54,55]. As shown in Figure 1, reduced ions should stimulate crystalline nuclei first. Then, depositions growth of multiple nuclear points on the entire electrode surface. Calculating the current transient of the growth of a single nuclear point first is necessary to investigate the nucleation, growth and current variation in an entire electrode surface over time. According to the microscopic characteristics of electrocrystallisation growth, the supply region of ion diffusion and migration growing around a single nuclear point should be hemispherical or cylindrical [26,56] when the current time function of a single nuclear point is calculated. The diffusion and migration radius of a single core point is set in the hemispherical diffusion mass transfer process, as indicated in Equation (8).

$$R={\left(kDt\right)}^{1/2}$$

(8)

By substituting Equation (7) with Equation (8), the diffusion current for a single core point can be obtained, as shown in Equation (9) [23,24].

$$i_{nuclei}=nF\pi {\left(2D{c}_0\right)}^{3/2}\times (M/\rho {)}^{1/2}\times t^{1/2}$$

(9)

where $i_{nuclei}$ is the current of growth of a single hemispherical nucleus, $M$ is the molecular weight of the deposited material, $\rho$ is the density of the deposited material, and $c_0$ is the ion concentration, respectively.

By performing the above calculation, the variation law of current generated by nucleation growth of a single point over time can be obtained. Studying the distribution and overlap of multiple nuclei on an electrode surface is necessary to obtain the change in crystal growth current over time. $N$ electrodeposition crystal growth nuclei are assumed to be present per unit area on an electrode surface. Avrami’s theorem describes the interaction between the nuclei and diffusion-influenced region, which has a time-dependent diffusion-influenced region radius, $R$.

Given the distribution of multiple nuclear points on an electrode surface and the overlapping effect of a diffusion-influenced region, the electrode surface coverage per unit area can be calculated using Equation (10) [25].

$$\theta =1-exp(-{\theta }_{ex})$$

(10)

where $\theta$ is the actual fraction of the area covered and ${\theta }_{ex}=N\times \pi R^2$ is the fraction of the area covered by diffusion zones without taking overlap into consideration, respectively.

Assuming that $N$ nuclear points grow on the electrode surface per unit area, Avrami’s theorem is used to convert the overlap of multiple proliferation mass transfer regions on an electrode surface to obtain the change function of the entire electrode current density over time, as shown in Equation (11) [57].

$$i_{density}=nF{c}_0{\left(D/\pi t\right)}^{1/2}\times (1-exp(-N\pi kDt\left)\right)$$

(11)

where $i_{density}$ is the actual current density on the electrode surface. Equation (11) directly establishes the relationship between the number, distribution and overlap of microscopic nuclear points and the macroscopic current on an electrode surface during crystal growth. This theory provides direct tools and a detailed theoretical basis for researchers to study the microscopic dynamic process of electrodes by using the current transient of electrode surfaces. It is extremely important for the dynamic study of electrode process and has been widely applied to electroplating research, such as Al [58], Bi [59], Cu [60,61,62,63], CuIn_xGa_1−xSe₂ [64], Li [65,66], Mg [67], Na [68], Ni [69,70,71], Ag [72,73,74,75], Pb [76,77], Pb-Pt Alloy [78], Pt [79,80], Si [81], WO₃ [82], Zn [83,84,85], Au [86], Ni-Se-Cu Alloy [87] and Au-Co Alloy [88,89].

However, certain deviations between the theoretical model [57] described by Equation (11) and the actual test curves were reported in experimental works [90,91]. In most literature reports, the experimental testing curves and theoretical model curves obtained from Equation (11) agree well before the time value corresponding to the current value peak. But there is an obvious deviation between the two curves after the time corresponding to the current density value peak. More importantly, the deviation between the experimental testing curves and theoretical model curves obtained from Equation (11) is gradually expands with prolonged electrodepositing time. This phenomenon was found in a large number of works during experimental current-transient curve testing. To be more precise, the attenuation speed of the experimental current-transient curve is slower than the curve obtained from Equation (11), as shown in Figure 3 in Ref. [91], Figure 4 in Ref. [92], Figure 4 in Ref. [90] and Figure 7 in Ref. [86].

Since then, a lot of work has been carried out to improve the electrodeposition crystallisation model. Among them, the more representative studies are discussed as follows. In 1999, D’Ajello et al. [90,93,94] developed a model to explain that the descending region of the i–t curve was not exactly consistent with the non-dimensional plots in Equation (11), as shown in Figure 8 in Ref. [26]. Ions with a solvatation sphere migrate describing a Brownian motion in the solution bulk was assumed in the work [90]. Further, ions randomly crossing a hemispherical surface, defined by a radius R, will lose their solvation shells and migrate towards an electroactive site by performing a diffusive motion assisted by an electric field, in the hemispherical area [90]. They also suggested current transients assuming that $N$ nuclear points grow on the electrode surface per unit area, as in Equation (12).

$$j=2\pi NDRzF{c}_{0}b\left(1+\frac{R{N}^{1/2}}{\sqrt{at}}\right)\left[1-exp\left(-at\right)\right]$$

(12)

where b is an appropriate proportional constant, and N is the nuclei density (nuclei per unit area), a = 4πND.

By considering the studies of D’Ajello and Scharifker together, Luo et al. [26] introduced a new limited-diffusion theory model. The model is based on a single hemispherical nucleus process revealed in Equations (11) and (12). The model described that the diffusion zone is limited, considering that electric field distribution on the electrode surface is limited. It gives a more detailed description of multiple nucleation processes during electrodepositing. The work introduced a new diffusion depth parameter, ‘R’ (the limited distance of the diffusion zone in the electrolyte) [26]. A new finite diffusion-controlled deposit growth-and-nucleation model based on diffusion migration and mass-transfer depth extension with stability tendency was established. In this new electrocrystallisation model, the ion random distribution area and the overlapping diffusion regions on the electrode surface of multiple-discharge deposited nucleus points (N) are still described by Avrami’s theorem. The current-density function of the N points discharge deposition process is established as Equation (13).

$$I=zF{c}_0{D}^{1/2}/{\pi }^{1/2}\left(\frac{\sqrt{\pi D}}{R}+\frac{1}{\sqrt{t}}\right)\left(1-exp\left(-N\pi kDt\right)\right)$$

(13)

Due to the introduction of the diffusion depth parameter ‘R’, compared to the mathematical relationship in Equation (11), the decay rate of the current-transient function in Equation (13) can be adjusted. Equation (11) is a limiting case of Equation (13), i.e., when parameter ‘R→∞’, expressions (13) and the current-density expressions (11) have the same expression. In addition, Milchev et al. [15,16,95], based on the theoretical and experimental research on the surface nucleation and growth process of micro and nano clusters in the electrochemical system, the deposition mechanism of micro and nano clusters was clarified, and relevant scientific theories were integrated in terms of thermodynamics and kinetics in the electrochemical reaction, and the deposition phenomena were explained and analysed. However, the microstructure and properties of crystalline layers in the electrochemical phase formation processes strongly depend on nucleation and growth. Native substrates do not require nucleation, and the growth method is influenced by the substrate’s perfection and the overpotential or supersaturation. Theoretical modeling of electrochemical nucleation and growth of metal nanocluster on a nanoelectrode was established by Isaev et al. [33,96]. General theoretical time dependences of the current and the nanocluster size have been presented for the case of diffusion-controlled growth at potentiostatic, galvanostatic and cyclic voltammetry deposition were noted in the study.

The study of the current transition function in electrocrystalline growth has been an important topic in electrochemical research for a long time [97,98]. Deeply studying the electrochemical microdynamics process is necessary to establish the accurate and reliable current transition function of electrocrystallisation. The systematic revelation of the microkinetic process of electrocrystallisation and the accurate calculation of the current transition function are the cornerstones of fine electrochemical machining and manufacturing technologies.

Based on the analysis above, it can be seen that there are two main methods to establish the current transient of electrochemical phase formation. One is based on the thermodynamic theory of electrodeposition, and the other is based on diffusion-controlled electrocrystallisation, as shown in Figure 2.

Figure 2a–c describe the process in which the Arrhenius formula is used to show the reaction rate. The positive and negative electrode reaction rates are obtained, and the current transient of electrochemical phase formation on the entire electrode surface is obtained by summing the two half-reactions. Figure 2d–g reflect the current transient of electrochemical phase formation controlled by ion transport and mass transfer, including the diffusion current on the ideal electrode surface (Cottrell equation), the current of a single nucleus, the mathematical model of multiple nuclei taking diffusion region overlapping into consideration, and the current transient establishment when multiple nuclear points grow together on the electrode surface. These are the two most important methods to guide the establishment of the current transient of electrochemical phase formation.

5. Conclusions

The microreaction kinetic process of electrocrystallisation determines the electrochemical phase formation and properties of materials. The study of electrochemical phase formation processes by in situ electrochemical methods utilising current transient to reveal the microcrystallisation kinetic process has certain advantages over other methods in the investigation of heterogeneous nucleation. Electrochemical phase formation plays an important role in the study of electrochemical basic theory and electrochemical fabrication techniques particularly for generating nanoelectronic devices and in the development of novel materials with unconventional properties, such as LECD. The shortcomings of existing studies were pointed out, and recommendations for improvement were proposed. As far as we know from the investigation: (1) there is no theoretical model that observed the nucleation and growth process during electrocrystallisation, directly. Therefore, it is very important to verify the validity and reliability of the theoretical model by using the experimental data obtained from the actual nucleation and growth process detection on the electrode surface. (2) The transient current functions established in various reported theoretical works are complex and difficult to use. It is of great practical significance to establish more useful current transient functions or transform the application process of complex formulas into a reasonable mathematical transformation, to make theoretical calculations and current measurements convenient. (3) The correlations among nucleation, growth and microstructure of deposits have not been clarified yet. That is, these correlations are useful in gaining deep insight into the preparation of microstructures. (4) In addition, three-dimensional nano-printing based on LECD technology has demonstrated strong potential for nano-scale metal microstructures manufacturing. It will be appealing to guide three-dimensional nano-printing equipment fabricating high-precision nano-metal devices, utilizing electrocrystallisation nucleation and growth theory.