Measurements of the anodic and cathodic diffusion-limited current densities *i*_{lCl}- and *i*_{lCuCl2}-.

## 1. Introduction

For understanding the mechanism of electrolysis it is important to estimate kinetics parameters controlling the atomistic reaction process of metal electrode that is polarized in an electrolyte solution, but it seems not to have been performed satisfactorily. The reason for this is attributed to the fact that because actual electrode reactions proceed quasi-reversibly via consecutive two processes which consist of surface reaction and volume diffusion of ions involved in the reaction, the expression for its current density/overpotential relationship have become complex and not been presented explicitly. This is also related to the subjects of studies concerning the process of deposition or dissolution of atoms in crystal growth or its dissolution.

It is well known that the etch pits having a crystallographic symmetry are formed at dislocation sites of the low indices surfaces of a crystal which was etched under a specified condition (e.g. Gilman et al., 1958; Young, Jr., 1961). The dislocation etch pit is thought to be formed via a nucleation and growth process of two-dimensional pits at the dislocation site or via a spiral dissolution of the surface step which is caused by screw dislocation (Burton et al., 1951; Cabrela and Levine, 1956). Therefore elucidation of its formation mechanism is important for understanding of the surface step motion which is thought to play major role in the dissolution process of a crystal, and dissolution kinetics of crystals in the etch pit formation has been investigated and discussed by some researchers (e.g. Ives and Hirth, 1960; Schaarwächter, 1965; Jasper and Schaarwächter,1966; Van Der Hoek et al., 1983) so far.

However the research concerning parameters controlling surface step motion in the dissolution of crystals has not been satisfactorily performed. Especially it has not been examined quantitatively except for a few studies (e.g. Onuma, 1991). This is principally due to the reason that because the dissolution of a crystal proceeds generally via a dissolution reaction of surface atom and diffusion process of the dissolved atom (ion) into interior of solution, it is difficult to experimentally inspect the dissolution kinetics of surface step which depends on both processes. Since the dissolution rate of a metal crystal which is anodically dissolved under polarization in an electrolyte solution can be investigated by measurement of current density, dissolution mechanism of metal crystals has been researched electrochemically (e.g. Despic and Bockris, 1960; Lee and Nobe, 1986). However because of the same reason as the above mention, discussions on the results have become complex and not always contributed to understanding of surface step motion.

Recently, however, it has been proposed by the author that an expression to analyze the relationship between anodic current density and overpotential of a quasi-reversible electrode system including both the consecutive reaction processes is derived explicitly on the basis of an appropriate assumption (Imashimizu, 2010, 2011). According to the analysis, if the anodic and cathodic diffusion-limited current densities are measured for a given quasi-reversible electrode system, we can experimentally determine the kinetics parameters controlling dissolution process of crystals of the metal electrode, by assuming expressions for the activation and concentration overpotentials which are driving forces of surface reaction process and volume diffusion process respectively.

Thus the dissolution rates at dislocation-free and edge dislocation sites of (111) surface when a copper crystal was anodically dissolved in an electrolyte solution are investigated and discussed based on the above thinking, in this chapter. The relationships between anodic current density and overpotential are analyzed and discussed electrochemically by using the method developed for anodic dissolution processes of quasi-reversible electrode as described above. Activation enthalpy, transfer coefficient and surface concentrations of the ions involved in the dissolution process are experimentally estimated, and kinetics parameters controlling anodic reaction of the copper crystal/electrolyte system are quantitatively examined. An expression for the vertical dissolution rate at dislocation site is proposed based on a nucleation model of two-dimensional pit, and the critical free energy change at nucleation is quantitatively examined.

## 2. Experimental procedures for study of dissolution kinetics of copper crystals

### 2.1. Preparation of specimens

Single crystals of copper with [111] direction about 10 mm in diameter were prepared from the starting material of re-electrolyzed copper of 99.996 % purity by using the pulling method. They were divided into the cylindrical crystals approximately 15 mm length by a strain-free cutting. A terminal for detection of electric current and potential was soldered to an end surface of the cylindrical crystals. Another end surface was chemically polished so that the deviation of surface orientation from [111] direction is within 8.7×10^{-3} rad, and was further electrolytically polished in a high concentrated phosphoric acid solution. The crystal specimen was embedded in a Teflon holder with paraffin so that its polished surface is exposed. Then it was supplied for the electrolysis experiment after the boundary portion between the paraffin and the periphery of polished surface was covered with a vinyl seal having a hole 6 mm in diameter (Watanabé et al., 2003).

### 2.2. Apparatus for potentiostatic electrolysis

Schematic diagram of the electrolytic cell for this experiment is shown in Fig.1 (Watanabé et al., 2003). The crystal specimen was immersed in the electrolyte solution which consists of 5 kmol m^{-3} NaCl, 0.25 kmol m^{-3} NaBr and 10^{-4} kmol m^{-3} CuCl (Jasper and Schaarwächter, 1966) so that (111) surface of copper is located at approximately 5mm below the surface of electrolyte solution, and was held at a specified temperature. Then, the crystal specimen was set under a constant overpotential, and (111) surface of the crystal was anodically dissolved for a prescribed time, while anodic current density/time curve was recorded. The potentiostatic electrolysis experiments were performed at a range of lower overpotential and at a range of higher overpotential. After that the structure of dissolved surfaces were observed by use of the optical microscope system equipped with lens for interferometry.

### 2.3. Features of anodic current density/time curves and structure of dissolved surface

Figure 2 shows typical anodic current density/time curves which were recorded while the copper crystal was anodically dissolved for 360 s or 600 s at the respective overpotentials. Anodic current density under any condition decreases steeply immediately after start of electrolysis and reaches a nearly constant current density *i*_{s} when it was carried out at an overpotential lower than about 125 mV as shown by the curve of 87 mV in Fig.2. Figures 3 (a), (b) and (c) are the optical micrographs of the (111) surfaces which were dissolved for 600 s at overpotentials in a range of 60 mV to 125 mV being held at 298K. The surfaces are rather smooth though etch pits tend to be formed as overpotential increases.

On the other hand, the current density reaches a minimum current density *i*_{sm} that is pointed by arrow after the initial steep decrease when an overpotential higher than about 125 mV was applied. Then it tends to increase gradually along with fluctuating and take a higher steady value as shown by the curve of 176 mV. Figures 4 (a), (b), and (c) are optical micrographs of the surfaces which were dissolved for 300 s at 156, 166, and 176 mV respectively being held at 298K. One can see that etch pits are significantly formed.

### 2.4. Measurement of anodic current density

*2.4.1. Steady anodic current densities at lower overpotentials*

The initial steep decrease of current density is principally due to the fact that a diffusion layer of the dissolved atoms (ions) forms in the neighborhood of crystal surface in process of time so as to decrease the undersaturation which is driving force for the dissolution. Therefore an approximately constant current density after its initial steep decrease is thought to be a steady current density *i*_{s} which flows accompanying with the consecutive two dissolution processes consisting of surface reaction and volume diffusion of dissolved atoms. This shows that the copper crystal/electrolyte system is a quasi-reversible electrode. Also the steady anodic current densities at lower overpotentials are thought to have only a little influence of formation of dislocation etch pits. Thus we assume that *i*_{s} is related to the vertical dissolution rate *v*_{s} at dislocation-free site of surface, which is given by the following expression (Schaarwächter and Lücke, 1967):

where *e* [C] is elementary charge (electronic charge), *n* [1] the charge number transferred at reaction and *Ω* [m^{3}] the atomic volume.

In this experiment, the potentiostatic electrolysis at overpotentials in a range from about 60 mV to 125 mV were carried out for 600 s at each temperature of 268, 283, 298 and 308 K and the relationships between the steady anodic current densities *i*_{s} and applied overpotentials *η* were investigated.

*2.4.2. Minimum anodic current densities at higher overpotentials*

On the other hand, the current density reaches a minimum after an initial steep decrease as shown in the curve of 176 mV in Fig.2 when an overpotential higher than about 125mV was applied. Then it tends to increase gradually together with fluctuating and take a higher steady value as described in Section 2.3. This is thought to be due to the fact that etch pits remarkably formed at dislocation sites and grew along with time under higher overpotential as shown in Fig.4 (Schaarwächter and Lücke, 1967; Imashimizu and Watanabé, 1983). That is, it is because nucleation and growth of etch pits at dislocation site resulted in an increase of the anodic current density which represents an average dissolution rate of whole surface exposed to electrolyte solution as the areas occupied by etch pits increase. Based on the above knowledge, we assume that the initial minimum current density *i*_{sm} under potentiostatic electrolysis at higher overpotentials is approximately equal to a current density that is equivalent to the dissolution rate of dislocation-free site of surface because the contribution to anodic current density of dislocation etch pit formation is thought to be a little in the initial stage of electrolysis. That is, an average value of *i*_{sm} was assumed to give the vertical dissolution rate at dislocation-free site of surface approximately as represented by the relation:

Thus the electrolysis experiment was carried out for a prescribed time from 60 s to 360 s at each overpotential of 156, 166, 176 and 186 mV keeping the temperature at 298 K and at each temperature of 268, 283, 298 and 308 K under an overpotential of 176 mV. The initial minimum current densities *i*_{sm} were obtained from the anodic current density/time curves measured under every condition.

### 2.5. Measurement of polarization curve and estimation of the diffusion-limited current densities

It needs to estimate activation overpotential *η*_{a} and concentration overpotential *η*_{c} for analyzing the relationship between anodic current density and applied overpotential as described in Section 1. Thus the polarization curves in a range of overpotential of about -400 mV to 400 mV were measured three times at each temperature of 298K and 308K by the potential step method. Then the anodic and cathodic diffusion-limited current densities were estimated.

### 2.6. Direct measurement of dissolution rates of surface

*2.6.1. Vertical dissolution rate of surface*

After the (111) surface of a copper crystal specimen was anodically dissolved at every condition of specified overpotrntials and temperatures as described in Section 2.4.2, it was observed by use of the optical microscope equipped with objective lenses for two-beam interferometry and multiple interferometry.

Figure 5 (a) shows the micrograph of a part of boundary region between the crystal surface exposed to the electrolyte solution and the peripheral portion covered with vinyl seal, which was photographed with two-beam interferometry mode. The vertical dissolution amounts *s* of surface shown by the illustration was measured from a deviation of the interference stripes caused by the step which was formed at that boundary region after dissolved. The vertical dissolution amounts *s* of surface under each condition was plotted against dissolution time *t*. The increasing rate *s* with *t* was obtained from the gradient of each linear relationship, and the vertical dissolution rate of surface under every condition was estimated by the

*2.6.2. Dissolution rates at dislocation site of surface*

Figures 5 (b) and (c) show a pair of micrographs of identical dislocation etch pits formed on dissolved surface which were photographed with optical mode and multiple interferometry mode. In this work, the depth *d* of the dark (deep) pits that were formed at positive edge dislocation sites (see Appendix A1) were measured by drawing the vertical cross sections of the pits that is shown by the illustration with use of the micrograph pairs such as Figs.5 (b) and (c). Also the width *w* (average distance from center to the three sides of pit) of those dark pits that is shown by the illustration were measured on the micrograph such as Fig.5 (b). Measurements of the depth and width of pit were performed about more than 20 dark pits formed on the surface dissolved under every condition, and the respective average values *d* and *w* were obtained. The depth *d* and the width *w* of dark pits were plotted against dissolution time *t*.

The increasing rate *d* with *t* was obtained from the gradient of each linear relationship, and the vertical dissolution rate *v*_{ed} at edge dislocation site was estimated by

where *v*_{s} means the vertical dissolution rate at dislocation-free site of the surface. Also the increasing rate *w* with *t* was obtained from the gradient of each linear relationship, and the lateral dissolution rate *v*_{w} at edge dislocation site was estimated from the relation:

### 2.7. Analysis of relationship between current density and overpotential

Under potentiostatic electrolysis of the copper/electrolyte system in the present experiment, the copper crystal is thought to be dissolved accompanying an anodic current according to a simple electrode reaction expressed by the following equation (Lal and Thirsk, 1953; Jasper and Scaarwächter, 1966):

where the contribution to current density of reaction of Br^{-} ion involved in dissolution process as inhibitor is assumed to be disregarded. The anodic current density *i*_{s} flowing steadily at an applied overpotential *η* is generally expressed by a relation:

where exchange current density *i*_{0} is represented by

(Tamamushi, 1967; Maeda 1961). *k*_{s} is surface density of kink that is active site at dissolution of surface atom, *α* the transfer coefficient, Δ*H*_{0} the activation energy (enthalpy) at dissolution of an atom, *ν* the atomic frequency, and *β* a supplementary factor of rate constant of electrode reaction. Also, *C*_{Cl}- and *C*_{CuCl2}- are the surface concentrations of Cl^{-} and CuCl_{2}^{-} ions involved in a steady anodic dissolution, and *C*^{0}_{Cl}- and *C*^{0}_{CuCl2}- the ones in equilibrium state. They are represented as a relative surface density as follows.

If the electrolyte solution contacting with crystal surface contained X ions of *m* kmol m^{-3}, the surface concentration *C*_{X} [1] can be expressed by the following relation:

where *N*_{A} is the Avogadro constant, *bb** the area occupied by an atom and *ξ* the thickness of electrolyte solution layer contacting with the crystal surface (Imashimizu, 2011).

The anodic dissolution of copper crystal in this experiment is thought to proceed quasi-reversibly with a surface reaction and volume diffusion of dissolution atom as described in Section 2.4.1. So we assume that the activation overpotential *η*_{a} and the concentration overpotential *η*_{c} are written by

where the folloing relations:

are given, if *i*_{lCl}- and *i*_{lCuCl2}- are the anodic and cathodic diffusion-limited current densities of the electrode reaction respectively (Tamamushi, 1967). Thus activation overpotential *η*_{a} and concentration overpotential *η*_{c} are assumed to be given by Eqs.(9) and (10), when the anodic dissolution of copper crystal proceeds steadily at an applied overpotential *η* by a quasi-reversible electrode reaction of Eq.(5). Also surface undersaturation *σ* is defined by

Then, the Eq.(6) is reduced to

by using Eqs. (7), (9) and (11). Also Eq.(12) leads to the following relation:

Thus if the anodic and cathodic diffusion-limited current densities *i*_{lCl}^{−} and *i*_{lCuCl2}^{-} are obtained, the experimental relationship of *i*_{s}/*η* would be represented with use of Eqs.(10) and (11) by Eq.(13). Then *α* and *i*_{0}(*T*) would be estimated from the gradient and the constant term of the linear relationship of ln{*i*_{s}(*C*_{Cl}-/*C*^{0}_{Cl}-)^{-2}*σ* ^{-1}} vs. *neη*/*kT*. Also Δ*H*_{0} would be estimated from the gradient of the linear relationship of ln{*i*_{0}(*T*)} vs. 1/*T*.

On the other hand, concerning the complex term consisting of surface concentrations of Cl^{−} and CuCl_{2}^{−} ions,

is lead from Eq.(9). Therefore applying Eq.(14) to Eq.(12) lead to

where Δ*H* is given by the relation:

We can see that Eqs.(15) and (16) are formulae for the steady current density expressed with use of the parameters *β*, *k*_{s}, *C*_{Cl}-, *C*_{CuCl2}-, *σ*, *α* and Δ*H* involved in the surface reaction process when the anodic dissolution progresses steadily by a quasi-reversible electrode reaction.

Thus undersaturation *σ*, transfer coefficient *α* and activation enthalpy Δ*H*_{0} for the anodic dissolution reaction of copper crystal/electrolyte system will be estimated from experimental results, and a supplementary factor *β* and kink density *k*_{s} will be examined by a model of crystal dissolution in this study.

## 3. Experimental results

### 3.1. Polarization curves and undersaturation in anodic dissolution

The polarization characteristic of the copper crystal/electrolyte system at 298K is shown in Fig. 6. The anodic and cathodic diffusion-limited current densities *i*_{lCl}- and *i*_{lCuCl2}- shown in the diagram were obtained by averaging the values measured three times. Table 1 shows those diffusion-limited current densities obtained from the polarization characteristics measured at 298 and 308 K by a similar method.

T /K | i_{lCl}-/A m^{-2} | i_{lCuCl2}-/ A m^{-2} |

298 | 827 | -0.0732 |

308 | 1072 | -0.156 |

The undersaturation *σ* were estimated from experimental polarization characteristics such as Fig.6 with use of Eqs.(10) and (11). The diagram that plotted *σ* against *neη*/*kT* in a range of (*neη*/*kT*) about 0 to 6 is shown in Fig.7. The black dots in the diagram show the values of *σ* which are calculated from the (*i*_{s}/*i*_{lCl}-)/(*neη*/*kT*) relationship that was derived by substituting the experimental values *i*_{0}, *α*, *i*_{lCl} and *i*_{lCuCl2} into Eq.(6).

The experimental relationships of *σ*/(*neη*/*kT*) at 298K and 308K approximately consist with each other, and also with the calculated relationship. However, the experimental curves of *σ*/(*neη*/*kT*) deviate from the calculated curve in a range of (*neη*/*kT*) larger than about 5. This is because the experimental current density includes an increase of current density attributed to significant formation of etch pits at higher overpotentials than about 125 mV. We assumed that the *σ*/(*neη*/*kT*) relationship does not almost depend on temperature from the result of Fig.7.

### 3.2. Estimations of parameters controlling exchange current density

Figure 8 is the diagram that plotted the steady current densities against overpotentials lower than 127 mV which were measured at 268K, 283K, 298K and 308K. Figure 9(a) is the diagram that plotted ln(*i*_{s}*σ*^{−1}) obtained from Fig. 8 against *neη*/*kT* at every temperature taking account of (*C*_{Cl}-/*C*^{0}_{Cl}-)^{−2} ≈1. The linear relationships at every temperature in the diagram are drawn so that they have a same gradient given by averaging. The transfer coefficient *α* was estimated from the gradient of their linear relationships. Then also the exchange current densities *i*_{0}(*T*) at each temperature were estimated from the constant terms of them. Figure 9(b) is the diagram that plotted ln{*i*_{0}(*T*)} against 1/*T*. The activation enthalpy for the anodic dissolution reaction of copper crystals was estimated from the gradient of the linear relationship shown in Fig. 9(b).

Then, because surface concentrations *C*^{0}_{Cl}-and *C*^{0}_{CuCl2} of Cl^{−} and CuCl_{2}^{−} ions are calculated by Eq.(8) when (111) surface of a copper crystal is in equilibrium with the electrolyte solution consisting of 5 kmol m^{-3} NaCl and 10^{-4} kmol m^{-3} CuCl, the complex term *C*^{0}_{Cl}-^{2(1-α)}*C*^{0}_{CuCl2}-^{α} in Eq.(7) giving exchange current density can be evaluated by using the transfer coefficient *α* estimated above.

Thus the estimations of parameters controlling exchange current density are summarized in Table 2. The value of* βk*_{s} was evaluated by substituting i_{0}, ΔH_{0} and C^{0}_{Cl}-^{2(1-α)}C^{0}_{CuCl2}-^{α} into Eq.(7), where the atomic frequency ν = 6.21×10^{12} [s^{-1}], elementary electric charge, e = 1.602× 10^{-19}[C] and n = 1 were assumed.

α | ΔH_{0} / eV | i_{0} /10^{-2}A m^{-2} | C^{0}Cl-^{2(1-α)}C^{0}_{CuCl2}-^{α} | β ks /10^{16} m^{-2} |

0.84 | 0.33 | 8.2* | 2.36×10^{-6} | 1.32 |

### 3.3. Anodic dissolution rates at higher overpotentials

#### 3.3.1. Estimation of vertical dissolution rate of surface from anodic current density

Figures 10 (a) and (b) are examples of the anodic current density/time curves which were recorded when the copper crystal was dissolved for 240 s at higher overpotential. The vertical dissolution rate v_{sm} at dislocation-free site of surface under every condition was determined from an average of the initial minimum current densities i_{sm} pointed by arrow of i/t curves (measured for five different dissolution time in a range of 60 s to 360 s under each condition) shown in Fig.10 as described in Sections 2.4.2.

#### 3.3.2. Estimation of vertical dissolution rate of surface by direct measurement

Figures 11 (a) and (b) are the diagrams that plotted vertical dissolution amounts s of surface against dissolution time t as described in Sections 2.6.1. The vertical dissolution rates

#### 3.3.3. Estimation of dissolution rates at edge dislocation site by direct measurement

Figures 12 (a) and (b) are the diagram that plotted the depth d of the dark etch pits which were formed at positive edge dislocation sites on the surface dissolved under each condition against dissolution time t, as described in Sections 2.6.2. Also Figs.13 (a) and (b) are the diagrams that plotted similarly the width w of the same dark etch pits as the above mention against dissolution time t. The increasing rate

### 3.4. Effects on the dissolution rates of overpotential and temperature

Figures 14 (a) and (b) are the diagrams that plotted the logarithm values of the dissolution rates. It can be seen that the value of log v_{sm} approximately consists with that of log_{sm} and _{s} of dislocation-free site of surface approximately. However, the value of v_{sm} seems to be more exact than that of_{s} of dislocation-free surface was assumed to be given not by _{sm}. Then v_{ed} and v_{w} in Fig.14 show the values estimated from Eq.(3) and Eq.(4) in which v_{s} was substituted by v_{sm}.

It can be seen that both log v_{sm} and log v_{w} tend to increase rather homogeneously with an increase of η, from Fig.14 (a). However, the tendency of log v_{ed} are somewhat different and in accelerative. Also, it can be seen from Fig.14 (b) that though log v_{sm} and log v_{w} tend to similarly increase with an increase of T, the tendency of log v_{ed} are somewhat little, compared to the former two. This is seen from the fact that the increasing rate (Δlog v_{ed}/ΔT = 5.4×10^{-3}) of the latter is less than that (Δlog v_{sm}/ΔT = 1.1×10^{-2}, Δlog v_{w}/ΔT =1.4×10^{-2}) of the former two.

## 4. Discussion

### 4.1. Atomistic dissolution model of crystal surface

#### 4.1.1. Vertical dissolution rate at dislocation-free site of surface

Concerning the dissolution of a crystal, the atomistic model illustrated in schematic diagram of Fig.15 has been proposed (Burton et al., 1951; Schaarwächter, 1965). The dissolution of crystals proceeds via a lateral retreat motion of surface step of an atomic height that is induced by dissolving of surface atom from the kink sites into the solution. The vertical dissolution rate v_{s} of surface is given by lateral retreat rate v_{h} and surface density tanθ of surface step, which is expressed by the following equation:

where θ is an average inclination of crystal surface to a low index face, a an atomic height of surface step, and λ the mean distance between adjacent surface steps. The lateral retreat rate v_{h} of surface step is expressed by

where ΔH_{s} is the activation enthalpy for dissolution of an atom at kink site of surface step, ν the atomic frequency, k* the retreat rate constant of surface step, and b* the unit retreat distance. σ_{s} is surface undersaturation, which is written as

where Δμ is the chemical potential difference of dissolution atom between two phases of a crystal/ solution system (Schaarwächter, 1965).

#### 4.1.2. Lateral dissolution rate at edge dislocation site

The dislocation etch pit is thought to be formed via a successive nucleation and growth processes of two-dimensional pits at the dislocation site (Schaarwächter, 1965) or via a spiral dissolution of the surface step which is caused by screw dislocation (Cabrera and Levine, 1956). We discuss the dissolution rate at edge dislocation site of (111) surface of copper crystals, based on a nucleation and growth model of two-dimensional pits (Schaarwächter, 1965) that is illustrated in Fig 16, in the following.

Since the lateral dissolution rate v_{w} is thought to represent horizontal growth rate of two-dimensional pit nucleated at edge dislocation site of surface, it may be corresponding to the lateral retreat rate v_{h} of surface step along (111) face. Thus we assume that v_{h} is given by v_{w} as shown in the following relation:

#### 4.1.3. Vertical dissolution rate at edge dislocation site

On the other hand the vertical dissolution rate at positive edge dislocation site would be examined by the nucleation rate of two-dimensional pit at dislocation site as follows.

According to the classical nucleation theory, if ΔG_{ed}* is the critical free energy change at nucleation of a two-dimensional pit at edge dislocation site, a steady state nucleation rate I of two-dimensional pit would be expressed by

where r is a separation rate of an atom from an active site of the two-dimensional pit into the solution and Z the Zeldovich factor (Toschev, 1973). Since the separation rate r is assumed to be a similar quantity to the dissolution rate of an atom from kink site of surface, it depends on the surface concentrations of Cl^{−} and CuCl_{2}^{−} ions as known from the Eq. (15) in Section 2.7, and is expressed by

Accordingly, the vertical dissolution rate v_{ed} at edge dislocation site of surface is expressed by

where a is the depth of "two-dimensional pit and K_{s} is an undetermined constant including Zeldovich factor and others (see Appendix A2.).

According to the nucleation theory of dissolution of crystals, ΔG_{ed}* is small compared to ΔG_{s}* which is the critical free energy change at nucleation of a two-dimensional pit at dislocation-free site of surface, because of strain energy of dislocation core. It is expressed by

and

where γ is the interfacial free energy of the crystal and solution at step of the two-dimensional pit, G the shear modulus and q and α_{c} the constants (Schaarwächter, 1965).

### 4.2. Relations between vertical dissolution rate of surface and anodic current density

#### 4.2.1. Expression for dissolution rate of dislocation-free site of surface

When the copper crystal is anodically dissolved by the simple electrode reaction of Eq.(5) the vertical dissolution rate v_{s} of dislocation-free surface at lower overpotentials and the v_{sm} at higher overpotentials would be estimated by Eq.(1) and Eq.(2) respectively as described in Section 2.4. Thus it is experimentally estimated with use of Eqs. (1), (2), and (15) by the following expression:

According to the dissolution model of crystals, the dissolution rate at dislocation-free site of surface is expressed from Eqs. (17) and (18) by

Therefore, the following relations are obtained from Eqs.(16), (26) and (27) concerning the rate constant of the lateral retreat rate of surface step and activation enthalpy for the dissolution.

and

is obtained. It can be seen that the rate constant k* of lateral retreat motion of surface step is electrochemically expressed by Eq.(28) and that it increases with an increase of concentration overpotential η_{c}.

#### 4.2.2. Estimation of kinetics parameters controlling the dissolution rate

As mentioned above the dissolution rate v_{sm} at dislocation-free site of surface under higher overpotentials is expressed by an approximate equation:

from Eqs. (14) and (26), where we assumed i_{s} << i_{lCl}-, that is,

Thus concerning the dissolution rate at dislocation-free site of surface which have a constant kink density k_{s}, a following approximate expression is lead from Eq.(31) (Imashimizu, 2011).

Figures 17 (a) and (b) are the diagrams that plotted the dissolution rate v_{sm} shown in Figs.14 (a) and (b) on a natural logarithmic scale against η (T = 298K) and 1/T (η =176mV) respectively. It can be seen that the values of (αne/kT) and ((ΔH_{0} −αneη)/k) are estimated by comparing the Eq.(33) with gradients of the linear relationships drawn in Figs. 17 (a) and (b), because the overpotential and temperature dependences of σ in the range of 156 mV to 186 mV are assumed to be a little. Thus, α and ΔH_{0} were also obtained from overpotential dependence of vertical dissolution rate v_{sm} of surface at higher overpotentials and temperature dependence of that. The estimations are shown in Table 3, showing α and ΔH_{0} are in good agreement with those values in Table 2.

According to atomistic dissolution model of a crystal surface illustrated in Fig.15, the relation of θ = tan^{-1}(a/λ) = tan^{-1}(v_{s}/v_{h}) is lead from Eq. (17), which represents the inclination angle of surface to (111) face. Since it is approximately given by θ ≈ θ * = tan^{-1}(v_{sm}/v_{w}) with use of Eqs.(20) and (26), the values of θ* obtained from Fig.14 were plotted against η and T in Figs.18 (a) and (b). It can be seen that the tendencies of change in θ * against η and T are not clear and not reasonable. The average value of θ *_{av} is 2.1×10^{-2} rad, which is a little large compared to a deviation 8.7×10^{-3} rad from [111] direction that was aimed when we prepared the surface of specimen as described in Section 2.1. This is probably attributed to the fact that actual surface exposed to electrolyte solution was slightly spherical as a whole and was having microscopic swells. That is, the variation of their values seems to be due to experimental error. Thus the vertical dissolution rate at dislocation-free site of surface is assumed to be given by retreat rate of the surface steps which preexists on the prepared surface, which gives following relation:

Accordingly_{s} in Table 2 by using Eq. (34), which is shown in Table 3 where assumed θ* = θ*_{av} (0.021 rad).

α | ΔH_{0}/eV | v_{sm}/m s^{-1} | β b/x_{0} | v_{ed}/m s^{-1} | ΔG_{ed}*/eV | |

K_{s} = 1 | K_{s} = 0.2 | |||||

0.85 | 0.33 | 1.6×10^{-9†} | 0.034^{†} | 5.7×10^{-9†} | 0.16^{†} | 0.12^{†} |

### 4.3. Vertical dissolution rate at dislocation site

#### 4.3.1. Estimation of the critical free energy change for nucleation of two-dimensional pit

As mentioned in Section 4.1.2 the dissolution rate v_{ed} at edge dislocation site is expressed by Eq. (23), but if Eq. (14) is applied it is reduced to

Accordingly, if we assume C_{Cl}-/C^{0}_{Cl}- ≈ 1, ΔG_{ed}* is given by

Thus ΔG_{ed}* under each condition was estimated by Eq.(36) with use of experimental value of v_{ed} as well as estimations of α and ΔH_{0} which were obtained in Section 4.2.2. The ΔG_{ed}* estimated with use of two assumed values of undetermined constant K_{s} for a specified condition (η = 176 mV and T = 298K) are shown together with the values α and ΔH_{0} in Table 3, where a = 2.09×10^{-10} m and ν = 6.21×10^{12} s^{-1} were used.

According to the precedent theoretical study (Schaarwächter 1965), in which the conditions for the formation of visible etch pit at dislocation site were investigated on the basis of a proposed nucleation model, the critical free energy change is estimated to be 0.115 eV. The present estimation of ΔG_{ed}* approximately consists with that value as shown in Table 3, though the exact value of K_{s} can not be evaluated in this study. This is seemed to be reasonable as described in Appendix A2.

On the other hand, however, it was admitted that the value of ΔG_{ed}* varies with overpotential and temperature as mentioned below.

#### 4.3.2. Overpotential and temperature dependences of ΔG_{ed}*

Figures 19 (a) and (b) are the diagrams that plotted the square root of ΔG_{ed}* estimated assuming K_{s} = 1 by Eq. (36) against η and T respectively. It can be seen that ΔG_{ed}*^{1/2} is not constant but changes in different manners with increases in η and T. The reason for this is probably that ΔG_{ed}*^{1/2} is proportional to the interfacial energy γ as known from Eqs. (24) and (25).

It is known that the interfacial energy varies with electrode potential according to so-called electrocapillary curve (Tamamushi, 1967). Therefore, the change in ΔG_{ed}*^{1/2} with η is surmised to be due to the potential dependence of γ, because the overpotential dependences of the undersaturation σ and therefore that of Δμ = neη_{a} = −kTln(1-σ) in an overpotential range of 156 to 186 mV are assumed to be a little as described in Section 4.2.2. This is supported by the fact that Fig.19 (a) indicates a quadratic dependence similar to the electrocapillary curve. Also, it is inferred from Fig.19(b) and Eq.(24) that γ should increase with an increase in Τ, becauseΔμ tend to increase with increase in T. This is probably attributed to a decrease in specific adsorption of anion accompanied by an increase of interfacial energy with rising of temperature.

The overpotential dependence of log v_{ed} is in accelerative, and somewhat different from that of both log v_{sm} and log v_{w}. Also the increasing rate of log v_{ed} with increase in temperature is smaller than that of both log v_{sm} and log v_{w} as shown in Figs.14 (a) and (b). The reason for this seems to be attributed to the overpotential and temperature dependences of the interfacial energy of the electrode surface as mentioned above.

## 5. Conclusions

Following conclusions were obtained from the results and discussion:

The transfer coefficient, activation enthalpy and surface concentrations of the ions which control the dissolution reaction were estimated from measurements of the relationships between steady anodic current densities and applied overpotentials when copper crystals are dissolved in an electrolyte solution under potentiostatic electrolysis.

The values of a supplementary factor and kink density affecting rate constant of dissolution reaction were examined.

The dissolution rate at edge dislocation site of (111) surface of copper was discussed quantitatively by a nucleation model of two-dimensional pit based on the classical nucleation theory.

The present estimation of the critical free energy change ΔG

_{ed}* for nucleation of a two-dimensional pit at edge dislocation site reasonably consisted with the evaluation by the precedent study.The overpotential and temperature dependences of dissolution rate at edge dislocation site were somewhat different from those dependences of dissolution rate at dislocation-free site. The reason for this is probably that ΔG

_{ed}* changes according to the overpotential and temperature dependences of interfacial energy.

## Appendix

## A1. Kinds of dislocation etch pits and their characters

The surface of copper specimen on which some small glass spheres 300 μm in diameter were dropped beforehand was anodically etched by the present method. Fig.20 (a) is an optical micrograph of dissolved surface in which Rosseta pattern composed of dark and light etch pits was formed at the portion that was hit by a small glass sphere. This proves that dark and light etch pits are formed at the sites of positive and negative edge dislocations respectively because the six arms of Rosseta pattern are composed of rows of a pair of positive and negative edge dislocations.

In another experiment, the surface of prepared copper specimen was anodically etched first by the present method, and a distribution of etch pits were observed by the optical microscope. Subsequently after electropolished the etched surface of specimen, the surface was etched for 10 s by a modified Young's etchant prepared by Marukawa (Marukawa, 1967), and the same portion as the previous portion was observed. Figs.20 (b) and (c) are a pair of optical micrographs of the surfaces etched by such two methods. It has been reported by Marukawa that the dark (deep) and light (shallow) pits are formed at screw dislocations and edge dislocations on the surface etched by the modified Young's etchant respectively. Accordingly it can be seen that the light etch pits are formed at the sites of screw dislocations on the surface that was anodically etched by the present method, by comparing the kinds of etch pits which are observed in these micrographs. Thus Table 4 is obtained concerning dislocation characters related to dark and light etch pits.

Etching | Edge dislcation | Screw dislcation | |

(positive) | (negative) | ||

Chemical^{†} | Light | Light | Dark |

Electrolytic^{††} | Dark | Light | Light |

In this work, the depth and width of the dark (deep) pits were measured to investigate the dissolution amounts at positive edge dislocation sites.

## A2. Estimation of undetermined constant K_{s}

As described in the Section 4.1.3, if the separation rate r of an atom at nucleation of two-dimensional pit is a quantity similar to the dissolution rate of an atom from kink site of surface, it would need to take account of supplementary factor β affecting the exchange current density as a parameter involved in the separation rate r. Then the dissolution rate v_{ed} at edge dislocation site derived from the nucleation rate Eq. (21) is represented afresh by

Thus, we assume that the undetermined constant K_{s} is approximately given by a relation:

We have assumed in the Section 2.7 that the exchange current density i_{0} is given by Eq. (7) for simplification, but to be exact i_{0} should be expressed with use of the activities of the ions involved in the electrode reaction instead of the concentrations. Also, transmission coefficient should be taken account of as pre-exponential factors in Eq. (7). Therefore it is generally hard to estimate β including some unknown factors. However, concerning β of the present electrode reaction, β (b/x_{0}) = 0.034 was estimated experimentally as shown in Table 3. Also it can be seen from an observation of etch pit by optical microscope that surface steps have a structure along a crystallographic direction of the crystal. Accordingly if (b/x_{0}) is assumed to be a quantity of 0.02 to 0.2, it would give an estimation of β = 0.17~1.7.

On the other hand, if we assume the free energy change ΔG_{ed} (j) for formation of a two-dimensional pit consisting of j vacancies at edge dislocation site, it is written as

Then the critical size j* of two-dimensional pit and the critical free energy change ΔG_{ed}*( j*) are given by

respectively. It can be seen that ΔG_{ed}*(j*) is expressed by the same relation as Eq. (24), and that the factor p’ has the same contents with Eq. (25), that is, p’ = p. Then, Zeldovich factor is expressed from the definition (Toschev, 1973) by

Accordingly, Z = 0.76 is estimated, if p = 0.18 (Schaarwächter 1965), Δμ = 0.027 eV(σ = 0.65) (Imashimizu, 2011), ΔG_{ed}* = 0.12 eV (Table 3) and kT = 0.0257 eV (T = 298K) are used.

Thus K_{s} = 0.13~1.3 is estimated from Eq. (38), which suggests the reasonability of the assumed value of K_{s} shown in Table 3.