ISSN 1023-1935, Russian Journal of Electrochemistry, 2008, Vol. 44, No. 4, pp. 459–469. © Pleiades Publishing, Ltd. 2008. Original Russian Text © V.M. Volgin, A.D. Davydov, 2008, published in Elektrokhimiya, 2008, Vol. 44, No. 4, pp. 496–507.
The Limiting Current Density of Copper Electrodeposition on Vertical Electrode under the Conditions of Electrolyte Natural Convection V. M. Volgina,z and A. D. Davydovb aTula
bFrumkin
State University, pr. Lenina 92, Tula, 300600 Russia Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninskii pr. 31, Moscow, 119991 Russia Received March 2, 2007
Abstract—The problem of determining the limiting current density of copper electrodeposition on the vertical electrode under the conditions of electrolyte natural convection is solved numerically. The incomplete dissociation of copper sulfate and sulfuric acid by the second stage and the concentration dependences of physicochemical properties of solution are taken into consideration. The effect of incomplete dissociation of copper sulfate and sulfuric acid by the second stage on the ionic transport in the copper electrodeposition and on the limiting current density is estimated by comparing the results, which are calculated taking into account and ignoring the concentration dependences of dissociation constants, and also for the limiting cases of virtually complete dissociation of copper sulfate and sulfuric acid. It is shown that, in excess of supporting electrolyte, incomplete dissociation of copper sulfate and sulfuric acid only slightly affects the limiting current density: the limiting current density changes not more than by 10% with the variation of the dissociation constants by several orders of magnitude. At the moderate and small concentrations of supporting electrolyte, incomplete dissociation should be taken into account, because the error of limiting current density can exceed 50%. Key words: natural convection, limiting current density, migration, copper sulfate, sulfuric acid DOI: 10.1134/S1023193508040125
INTRODUCTION When the electrochemical reactions proceed under the diffusion control, the liquid flow, which is associated with the variation in the concentration and, correspondingly, density of solution, can be observed near the electrode. This flow, which is called the natural convection, is caused by the buoyancy force that arises due to the difference between the solution densities near the electrode surface and in the bulk electrolyte. The problem of determining the limiting current on the vertical electrode was solved for the first time by Levich [1]. He obtained the approximate analytical equation for the limiting current density that qualitatively properly gave the dependence of the limiting current on the concentration of electroactive ions, the electrode length, viscosity, and diffusion coefficient. However, the theoretical results differed significantly from the experimental data due to ignoring the migration transfer of supporting electrolyte. In contrast to the case of forced convection of electrolyte, when, in excess of supporting electrolyte, the variations in the concentration of nonelectroactive ions have almost no effect on the limiting current, in the case z
Corresponding author:
[email protected] (V. M. Volgin).
under consideration, the variations in the concentration of supporting electrolyte have a pronounced effect on the solution density and, consequently, on the limiting current. In [2–6], the analytical solutions that account for the variations in the concentration of supporting electrolyte were obtained using the approximate methods. In [2–4], the migration of indifferent electrolyte was taken into consideration by introducing the transport numbers, considerably simplifying the solution. However, the use of transport numbers in the systems with a gradient of concentration is not substantiated [7]. The authors of [6] managed to solve the problem without resort to the transport numbers and, in contrast to the earlier studies, they took into account the effect of diffusion coefficients not only of electroactive Cu2+ ions, but also of other types of ions. Comprehensive analysis of theoretical and experimental data on the natural convection in the electrochemical systems is presented in the review [8]. The possibilities of analytical theoretical study of electrochemical systems under the conditions of natural convection are considerably limited by the interrelation between the ionic transfer and the hydrodynamic processes. Therefore, along with the approximate analytical methods, the numerical methods are used for calculating natural convection in the electrochemical systems [9, 10]. In contrast to the ana-
459
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VOLGIN, DAVYDOV
lytical methods, the numerical methods enable one to take into consideration rather easily the concentration dependences of physicochemical properties of solution and the homogeneous chemical reactions. The peculiarity of copper electrodeposition from the copper sulfate and sulfuric acid solution is incomplete dissociation of copper sulfate and sulfuric acid (by the second stage). For simplicity, commonly, only two limiting cases of sulfuric acid dissociation by the second stage are con(Cu2+,
H+,
2– S O4
sidered: the complete dissociation and ions are present in the solution) and the absence of dis– sociation (Cu2+, H+, and HS O 4 ions are present in the solution) [9]. Incomplete dissociation of copper sulfate was ignored in the analysis of ionic transfer under the conditions of natural convection. It is difficult to take into account the homogeneous chemical reactions in the copper electrodeposition, because the equilibrium constants of dissociation of sulfuric acid and copper sulfate depend on the activity coefficients of ions, which, in their turn, depend on the concentration of electrolyte. There are several methods for determining the coefficients of activity [11]. The equation of Debye–Hückel can be used to determine the activity coefficients only at low concentrations of copper sulfate and sulfuric acid. In the range of higher concentrations, the modified Davies equation [11] or the equations of Pitzer [12] can be used. The Pitzer’s equations enable one to obtain more precise coefficients of activity; however, they require the knowledge of a larger number of parameters, which are determined experimentally. The absence of some parameters, in particular, the coefficient that accounts for the interaction of ions in the solution with CuSO4(aq) molecules, in the Pitzer’s model restricts the accuracy of calculation of ion activity coefficients at high concentrations of solution [13]. In [14, 15], the modified Davies equation for the solutions with a high ionic strength was proposed. In this case, relative simplicity of calculating the activity coefficients with reasonable accuracy is provided. The effect of homogeneous chemical reactions in the aqueous solution of copper sulfate and sulfuric acid on the distribution of concentrations of ions was studied in [13, 16–18], including the case of copper electrodeposition on a rotating disk electrode [18]. Earlier, the homogeneous chemical reactions in the copper electrodeposition on the vertical electrode under the conditions of natural convection were ignored. In this work, the numerical method is proposed for calculating the limiting current density of copper electrodeposition from aqueous solution of copper sulfate and sulfuric acid. In the method, the homogeneous chemical reactions (incomplete dissociation of copper sulfate and sulfuric acid by the second stage) and the concentration dependences of physicochemical properties of solution are taken into consideration.
STATEMENT OF PROBLEM In the aqueous solution of copper sulfate and sulfuric acid, ions of four types are present: (1) Cu2+, (2) H+, 2–
–
(3) S O 4 , (4) HS O 4 and (5) CuSO4(aq). It is believed that copper sulfate partially dissociates yielding Cu2+ 2–
and S O 4 , ions, whereas sulfuric acid dissociates com–
pletely only by the first stage yielding H+ and HS O 4 ions and partially by the second stage yielding H+ and 2–
S O 4 ions. Thus, in the copper sulfate and sulfuric acid solution, two high-rate homogeneous chemical reactions proceed: CuSO 4 ( aq )
Cu
2+
+
–
HSO 4
2–
+ SO 4 ,
(1)
2–
H + SO 4 .
(2)
Within the framework of Boussinesq approximation, the mass transfer in the boundary layer near the surface of vertical electrode of height H, on the Cartesian coordinates with the origin of coordinates on the lowest electrode point, the abscissa axis directed along the electrode surface, and the ordinate axis directed normally to the electrode surface, is described by the following set of equations in the dimensionless form [7, 19, 20]: ∂V ∂V 1 -------- ⎛ V X --------X- + V Y --------X-⎞ Sc 1 ⎝ ∂ X ∂Y ⎠ ∂V = ----------2X- + Ra 1 [ ( 1 – C s ) + β ( C ab – C a ) ], ∂Y ∂V X ∂V Y --------- + --------- = 0, ∂X ∂Y 2
∂C ∂Φ ∂ – ---------2-k – z k ⎛ C k -------⎞ ⎝ ∂Y ⎠ ∂ Y ∂Y 2
(3)
D ∂C ∂C + ------1 ⎛ V X --------k + V Y --------k + σ k⎞ = 0, ⎠ ∂Y Dk ⎝ ∂X
4
∑z C k
k
= 0,
k=1
C1 C3 = K s C5 , C2 C3 = K a C4 .
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THE LIMITING CURRENT DENSITY OF COPPER ELECTRODEPOSITION
Here, X and Y are the dimensionless coordinates; VX and VY are dimensionless components of hydrodynamic velocity; Ck, Dk, and zk are dimensionless concentration, diffusion coefficient, and the charge of species of the k-th type, respectively; Ca and Cs are dimensionless concentrations of acid and salt, respectively; Φ is dimensionless electric potential; β = (∂ρ/∂ca)/(∂ρ/∂cs) is the coefficient (the ratio between the mass coefficients of acid ∂ρ/∂c and salt ∂ρ/∂cs); σk is the intensity of bulk source of ions of the k-th type, which is associated with the homogeneous chemical reaction; Ks and Ka are dimensionless equilibrium constants for reactions (1) and (2), respectively; and b is the subscript referring the values to the bulk electrolyte. The Schmidt number is determined by the diffusion coefficient of electroactive ion: ν Sc 1 = ------. D1
(4)
The Rayleigh number is determined by the concentration of copper sulfate in the bulk electrolyte: 3
gH c s ∂ρ Ra 1 = ----------------b ------- , νD 1 ρ ∂c s
(5)
where g is the gravitational acceleration, ν is the kinematic viscosity, and ρ is the electrolyte density. When passing to the dimensionless variables, the ratio D1/H was taken to be the unit velocity; c sb (the total concentration of copper in CuSO4 in the bulk electrolyte), to be the unit concentration; RT/F (here, R is the gas constant, T is the temperature, and F is the Faraday number), to be the unit potential; and the electrode height H, to be the unit length. The concentrations Cs and Ca can be expressed in terms of concentrations of ions and CuSO4(aq): Cs = C1 + C5 ,
C2 + C4 -. C a = ----------------2
K s0 -, K s = --------------c sb γ 1 γ 3
(7)
K a0 γ 4 -. K a = --------------c sb γ 2 γ 3
(8)
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Here, K s0 , and K a0 are the equilibrium constants at infinite dilution and γk is the activity coefficient of ions of the kth type. The values of K s0 and K a0 depend on the temperature, and the following equations were used to calculate them [13]: K s0 = 1000 exp ( – 21.9808/T – 5.3607 ), K a0 = 1000 exp ( 2825.2/T – 14.0321 ).
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(9)
In equation (7), it was assumed that the activity coefficient of CuSO4(aq) is unity. The activity coefficients of ions were determined by the modified Davies equation, which was proposed in [14, 15]: ( 0.2 – qI )I I 2 ln γ k = – Az k ------------------------- – -------------------------- . 1000 1 + ak B I Here, I = 0.5 c sb
∑
2 4 z C k=1 k k
(10)
is the ionic strength of
2
2 2F e 0 2F --------- are the temsolution; A = ---------------------------, and B = 3/2 εRT 8π ( εRT ) perature-dependent parameters; ak is the radius of ions of the k-th type; q is the parameter; e0 is the electron charge; and ε is the permittivity. The boundary conditions for equations (3) for the mode of the limiting current of copper deposition can be presented as follows:
VX
Y=0
= VY
C1
Y=0
C2 C3
(6)
Set of equations (3) involves the equations of pulse and mass conservation for incompressible viscous liquid, the equations of convective electrodiffusion of electrolyte components, the condition of electroneutrality, and the equations of equilibrium of homogeneous chemical reactions (1) and (2). From equations (1) and (2), it follows that σ1 = –σ5 = σs, σ2 = –σ4 = σa, σ3 = σs + σa. The equilibrium constants Ks and Ka depend on the concentrations of ions. The concentration dependences of Ks and Ka are taken into consideration, as it was proposed in [12], by using the following equations:
461
= 0,
Y=0
= 0, Y=0
C5
VX
∂C ∂Φ D 2 ⎛ ---------2 + z 2 C 2 -------⎞ ⎝ ∂Y ∂Y ⎠ ∂C ∂Φ + D 4 ⎛ ---------4 + z 4 C 4 -------⎞ ⎝ ∂Y ∂Y ⎠
∂C ∂Φ D 3 ⎛ ---------3 + z 3 C 3 -------⎞ ⎝ ∂Y ∂Y ⎠
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Y→∞
Y=0
;
Y=0
(11)
= 0; Y=0
∂C ∂Φ + D 4 ⎛ ---------4 + z 4 C 4 -------⎞ ⎝ ∂Y ∂Y ⎠ Y=0
∂C + D 5 ---------5 ∂Y Ck
= 0;
= 0,
Y=0
= K a C4
Y→∞
= 0; Y=0
= Ckb ,
Φ
Y ref
= 0,
Y=0
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VOLGIN, DAVYDOV
where Yref is an arbitrary prescribed point, in which the potential is taken to be zero. At the electrode surface, the boundary conditions (11) express the condition of adhesion of viscous liquid to the solid electrode surface, the complete consumption of electroactive ions (the limiting current), and zero fluxes of indifferent ions. The bulk concentrations of components are determined by the given bulk concentrations of copper sulfate and sulfuric acid taking into account the conditions of equilibrium of homogeneous reactions (1) and (2). As a result of solving set of equations (3), the limiting current density of copper deposition, which is averaged over the electrode’s length, should be determined: 1
i limav =
∫i
lim d X,
(12)
0
2FD 1 c s ∂C D ∂C where ilim = – --------------------b ⎛ ---------1 + ------5 ---------5⎞ H ⎝ ∂Y D 1 ∂Y ⎠
is the local Y=0
limiting current density. SELF-SIMILAR VARIABLES For the limiting-current mode, the equations of natural convection can be solved by introducing the selfsimilar variables: the coordinate η and the stream function ψ [20]. To prescribe the self-similar variables, various equations can be used [9, 19, 20]. In particular, the self-similar variables can be defined using the Grash of number or the Rayleigh number. For the electrochemical systems, which are characterized by high Schmidt numbers, it is advantageous to use the self-similar variables, which are defined using the Rayleigh number, because, in this case, in the self-similar equations of material balance of electrolyte components, a small factor at the highest derivative is absent. In this work, the self-similar variables are defined as follows: Ra 1/4 Y -, η = ⎛ ---------1⎞ -------⎝ 4 ⎠ X 1/4 Ra 1/4 3/4 ψ = 4 ⎛ ---------1⎞ X f ( η ), ⎝ 4 ⎠
(13)
where f(η) is a certain function of self-similar variable η. After transformation (13), the components of velocity take the following form: VX
Ra 1/2 1/2 = 4 ⎛ ---------1⎞ X f ', ⎝ 4 ⎠
_ 1/4 Ra 1/4 V Y = 4 ⎛ ---------1⎞ ( ηf ' – 3 f ) X . ⎝ 4 ⎠
The equation for the limiting current density in the self-similar variables is as follows: 8FD 1 c s Ra 1/4 D5 ⎞ ' i limav = --------------------b ⎛ ---------1⎞ ⎛ C '1 + ------C 3H ⎝ 4 ⎠ ⎝ D 1 5⎠
.
(15)
η=0
Introducing the Sherwood number, equation (15) can be presented in the following dimensionless form: 1/4
Sh av = K L Ra 1 .
(16)
i limav H - is the Sherwood number averHere, Shav = ------------------2FD 1 c sb aged over the cathode surface and KL = 3/4 D5 ⎞ 4 ' -------- ⎛ C '1 + ------C 3 ⎝ D 1 5⎠
is the mass-transfer coefficient. η=0
After passing to the self-similar variables, the initial system of equations (3) is reduced to the system of ordinary differential equations for determining function f and the concentrations of ions: 1 2 f ''' + -------- [ 3 ff " – 2 ( f ' ) ] + 1 – C 1 – C 5 Sc 1 β + --- ( C 2b – C 2 + C 4b – C 4 ) = 0, 2 D1 D1 ' + ------Ω = 0, C "k z k C k Φ" + z k C 'k Φ' + 3 f ------C Dk k Dk k
4
∑z C k
k
= 0,
(17)
k=1
C1 C3 = K s C5 ,
C2 C3 = K a C4 ,
K s0 -, K s = --------------c sb γ 1 γ 3
K a0 γ 4 -, K a = --------------c sb γ 2 γ 3
( 0.2 – qI )I I 2 ln γ k = – Az k ------------------------- – -------------------------- , 1000 1 + ak B I 4
I = 0.5c sb
∑z C , 2 k
k
k=1
(14)
4 X 1/2 where Ωk = ⎛ -------⎞ σk. Since Ωk depends only on the ⎝ Ra⎠ concentrations of ions, system of equations (17) is self-
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THE LIMITING CURRENT DENSITY OF COPPER ELECTRODEPOSITION
similar. In the self-similar variables, the boundary conditions (11) take the following form: f
= f'
η=0
C1
η=0
C2 C3 D 2 ( C 2' + z 2 C 2 Φ' )
η→∞
C5
η→∞
η=0
= K a C4
η=0
= 0,
= 0,
η=0
,
+ D 4 ( C 4' + z 4 C 4 Φ' )
η=0
η=0
+ D 4 ( C '4 + z 4 C 4 Φ' )
+ D 5 C 5' Ck
= f'
= 0,
η=0
D 3 ( C '3 + z 3 C 3 Φ' )
η=0
Φ
η ref
to the required accuracy, the grid step near the electrode surface should be sufficiently small. Under these conditions, the use of nonuniform grid enables one to cut significantly the computational time providing the required accuracy. As a result of discretization of system of differential equations (17) and boundary conditions (18), taking into consideration equation (19) and the concentration dependence of the equilibrium constants of chemical reactions (1) and (2), we obtain the following system of difference equations:
η=0
1 2 2 δ V i + -------- ( 3 f i δV i – 2V i ) + 1 – C 1, i – C 5, i Sc 1
= 0,
η=0
= Ckb ,
= 0, (18)
463
β + --- ( C 2b – C 2, i + C 4b – C 4, i ) = 0, 2
= 0.
h i – 1 δV i h i – 1 δ V i f i = f i – 1 + h i – 1 V i – -----------------+ --------------------- , 6 2 3
2
NUMERICAL SOLUTION The numerical solving of system of equations (17) with boundary conditions (18) involve some difficulties, because the equations of the system are coupled and nonlinear, and the boundary conditions at η = 0 and η ∞ prevent the use of well-developed methods for solving the Cauchy problem for the systems of ordinary differential equations [21]. In addition, the approximation of boundary conditions for the third-order differential equation involves some problems. To simplify the numerical solution, a new variable is introduced: V = f'.
ηi + 1 = ηi + hi , h i = θh i – 1 .
2
2
D1 D1 + 3 f i ------δC k, i + ------Ω k, i = 0, Dk Dk 5
∑z C k
= 0,
k, i
k=1
C 1, i C 3, i = K s, i C 5, i ,
C 2, i C 3, i = K a, i C 4, i , K a0 γ 4, i -, K a, i = --------------------c sb γ 2, i γ 3, i
K s0 -, K s, i = --------------------c sb γ 1, i γ 3, i
Ii ( 0.2 – qI i )I i 2 - – ---------------------------- , ln γ k, i = – Az k ------------------------1 + ak B I i 1000 4
I i = 0.5c sb
∑z C 2 k
k, i ,
k=1
(20) f 0 = 0,
Here, hi is the grid step between the nodes with subscripts i and i + 1; θ is the coefficient of nonuniformity θ–1 - is the initial grid step; and N of grid step; h0 = L -------------N θ –1 is the number of grid nodes. At large Sc1 numbers, which are typical for the electrochemical systems, the thickness of diffusion boundary layer is much smaller than the hydrodynamic layer thickness; therefore, the dimension L of computational region should be sufficiently large. To calculate the distributions of concentrations of electrolyte components RUSSIAN JOURNAL OF ELECTROCHEMISTRY
δ C k, i + z k C k, i δ Φ i + z k δC k, i δΦ i
(19)
This enables us to obtain a set of nonlinear differential equations of the second order. For solving numerically system of equations (17) with boundary conditions (18), the finite-difference method is used. In the computational region 0 ≤ η ≤ L (L is the outer boundary of computational region), the nonuniform finite-difference grid with nodes ηi is constructed. The node η0 = 0 is located at the electrode surface, the coordinates of other grid nodes are determined as follows:
2
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V 0 = 0,
C 1, 0 = 0,
C 5, 0 = 0,
C 2, 0 C 3, 0 = K a0 C 4, 0 , D 2 ( δ C 2, 0 + z 2 C 2, 0 δ Φ 0 ) +
+
+ D 4 ( δ C 4, 0 + z 4 C 4, 0 δ Φ 0 ) = 0, +
+
D 3 ( δ C 3, 0 + z 3 C 3, 0 δ Φ 0 ) +
+
+ D 4 ( δ C 4, 0 + z 4 C 4, 0 δ Φ 0 ) + D 5 δ C 5, 0 = 0, +
V N = 0, No. 4
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+
Φ N = 0,
+
C k, N = C k b .
(21)
464
VOLGIN, DAVYDOV
Here, δAi and δ2Ai are the central-difference approximations to the first and second spatial derivatives of function A on the internal grid nodes, respectively; δ+A0 is the asymmetrical three-point approximation to the first spatial derivative of function A on the grid node, which is located on the electrode. The iterative solving of nonlinear system of difference equations (21) was performed using the Newton method [22]. To determine the limiting current density and the mass-transfer coefficient, the asymmetrical three-point approximations to the derivatives C '1 and C '5 . were used. RESULTS AND DISCUSSION According to the literature data, the following dependences of physicochemical properties of electrolyte were taken [22, 23]: ρ = KT ρ ( 997.05 + 0.1598c s + 0.063257c a – 0.00574c s – 0.00117c a ), 2
2
µ = 0.89 × 10 KT µ ( 1 + 0.5765c s + 0.17c a
(22)
–3
+ 0.2715c s + 0.022c a ), 2
2
(23)
where KTρ = 1 – 2.3 × 10–4(T – 298) and KTµ = 1 – 0.022(T – 298) are the temperature coefficients for the density and viscosity, respectively. To determine the diffusion coefficients, as well as in [24], it was assumed that ∞
Dk = K D Dk ,
(24)
∞
where D k is the diffusion coefficient of ions of the k-th type under the infinite dilution; KD = µ/µ∞ is the correction factor; and µ∞ is the viscosity under the infinite dilution. The following diffusion coefficients under the infinite dilution were taken [15]: ∞
∞
–9
∞
∞
–9
D 1 = D Cu2+ = 0.72 × 10 KT D1 m /s, 2
D 2 = D H+ = 9.312 × 10 KT D2 m /s, ∞
∞
2
D 3 = D SO2– = 1.065 × 10 KT D3 m /s, –9
2
4
∞
∞
D 4 = D HSO– = 1.33 × 10 KT D4 m /s, –9
2
4
∞
∞
D 5 = D CuSO4 ( aq ) = 0.72 × 10 KT D5 m /s, –9
2
where K T Dk is the temperature coefficient for the diffusion coefficient of ions of the k-th type. According to
the literature data [23], the following temperature coefficients were taken: KT D1 = 1 + 0.024 ( T – 298 ), KT D2 = 1 + 0.014 ( T – 298 ), KT D3 = KT D4 = 1 + 0.02 ( T – 298 ),
(25)
KT D5 = 1 + 0.024 ( T – 298 ). The mass coefficients were calculated using equation (22) at the concentrations of acid and salt in the bulk solution: ∂ρ ------- = KT ρ ( 0.1598 – 0.00574c sb ), ∂c s ∂ρ ------- = KT ρ ( 0.063257 – 2 × 0.00117c ab ). ∂c a
(26)
For the calculations, the following magnitudes of parameters were taken: L = 1000, h0 = 10–6, and N = 1000. These values provided the absence of the effect of finite dimensions of computational region on the calculated results and sufficient accuracy of calculated gradients of concentrations near the electrode surface. In order to estimate the effect of copper sulfate concentration and relative concentration of sulfuric acid, which was defined as follows: c ab -, r = ----------------c ab + c sb
(27)
the concentrations of ions and CuSO4(aq) in the bulk electrolyte were calculated using the modified Davies equation (10) (Fig. 1). Figure 1a gives the plots of dissociation degree of copper sulfate (the relative concentration of Cu2+ ions) vs. the relative concentration of sulfuric acid. At small concentrations of copper sulfate, its dissociation degree decreases with increasing r (Fig. 1, curves 1 and 2), whereas at high concentrations, it increases with increasing r (Fig. 1, curves 3–5). The character of dependences is in agreement with the data reported in [13] and is explained by two effects: a decrease in the dissociation degree with increasing concentration of copper sulfate and an increase in the dissociation degree with increasing concentration of sulfuric acid. At a constant r, the lower concentrations of copper sulfate correspond to the lower concentrations of sulfuric acid, leading to the predominance of the first effect. And, conversely, at the same r, the higher concentrations of copper sulfate correspond to the higher concentrations of sulfuric acid leading to the predominance of the second effect. In the range of concentrations under investigation, the dissociation of copper sulfate reaches the values below 0.7, indicating that incomplete dissociation of copper sulfate should be taken into consideration. The relative concentration of
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THE LIMITING CURRENT DENSITY OF COPPER ELECTRODEPOSITION
465
c4b c4b +c3b
clb csb
(b)
(a) 0.8
0.9
0.8 0.4 0.7 1 0.6 0.0
2 0.4
3
4
5 0.8
r
0.4
0.0
0.8
r
–
Fig. 1. Plots of relative concentration of (a) Cu2+ and (b) HS O 4 ions in the bulk electrolyte vs. the relative concentration of sulfuric acid, which are calculated using the modified Davies equation. Concentration c s , mol/m3: (1) 0.1, (2) 1, (3) 10, (4) 50, and (5) 100. b
–
HS O 4 ions in the bulk electrolyte monotonically increases with increasing r. This is explained by a decrease in the dissociation degree of bisulfate ions with increasing concentration of sulfuric acid [13]. The relative concentration of bisulfate ions varies over a wide range (from 0 to 1); therefore, for calculating the limiting current, incomplete dissociation of sulfuric acid by the second stage should be taken into account. Figure 2 gives the data that shows the effect of copper sulfate and sulfuric acid dissociation on the mass transfer near the vertical electrode. For the calculations, four cases were considered: (1) the equilibrium constants were calculated using the modified Davies equation; (2) the equilibrium constants were taken to be equal to the constants for the infinite dilution of electrolyte c sb Ks = K s0 = 10.50 and c sb Ka = K a0 = 4.36; (3) the equilibrium constant for sulfuric acid was taken to be equal to that for the infinite dilution of electrolyte c sb Ks = K s0 = 10.50, whereas the equilibrium constant for
of copper sulfate, which has an effect on the concentration of copper ions in the bulk electrolyte (Fig. 2a). At the complete dissociation of copper sulfate, the degree of sulfuric acid dissociation only slightly affects the distribution of copper ions over the diffusion layer (Fig. 2a, curves 3, 4, 7, and 8). The variation in the concentrations of nonelectroactive ions, which is caused by the different degrees of dissociation of copper sulfate and sulfuric acid (Figs. 2b–2e), has an effect on the magnitude and the character of variation of dimensionless vertical component of hydrodynamic velocity (Fig. 2e). The effect is most pronounced at small relative concentration of sulfuric acid (Fig. 2f, curves 1–4). The best agreement with the result calculated using the modified Davies equation is obtained under the assumption of complete dissociation of copper sulfate and sulfuric acid by the second stage. In excess of sulfuric acid (Fig. 2f, curves 5–8), the hydrodynamic velocity weakly depends on the method of calculation and the values of equilibrium constants and is lower, which is caused by an increase in the concentration of sulfuric acid near the cathode surface (Fig. 3).
copper sulfate was taken to be c sb Ka = 106, corresponding to virtually complete dissociation of copper sulfate; and (4) the equilibrium constants were taken to be c sb Ks = c sb Ka = 106, corresponding to the virtually complete dissociation of copper sulfate and sulfuric acid by the second stage.
version of calculation (Cu2+, H+, and S O 4 prevail in the solution) to –0.07 for the third version of calcula-
As is seen from this data (Figs. 2a–2e), the dissociation of copper sulfate and sulfuric acid has a pronounced effect on the distribution of ion concentrations over the diffusion layer. The distribution of electroactive ions depends primarily on the dissociation degree
tion (Cu2+, H+, and HS O 4 prevail in the solution). In the third version of calculation (virtually complete dissociation of copper sulfate and the absence of dissociation of sulfuric acid), the concentration of acid is lower near the cathode than in the bulk electrolyte (at sufficiently
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The relative increase in the concentration of acid near the electrode surface varies from 0.46 for the forth 2–
–
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466
VOLGIN, DAVYDOV
C1
C2–C2b Ia
Ib
0.8
1.0
0.4
0.5 1 2 3 4
5 6 7 8 0
0
IIa
IIb
0.8
1.0
0.4
0.5
0
4
2
η
0
2 2–
η
4 –
Fig. 2. Distributions of dimensionless concentrations of (a) Cu2+, (b) H+, (c) S O 4 , and (d) HS O 4 ions and (e) CuSO4(aq) and (f) distribution of vertical component of hydrodynamic velocity over the width of diffusion layer at c s of (I) 10 and (II) 100 mol/m3, b
which were calculated: (1 and 5) taking into account the concentration dependences of equilibrium constants Ks and Ka; (2 and 6) ignoring the concentration dependences of equilibrium constants ( c s Ks = K s and c s Ka = K a ); (3 and 7) c s Ks = 106 and b
0
b
0
b
c s Ka = K a ; (4 and 8) c s Ks = c s Ka = 106; (1–4) c a = 0.1 c s ; and (5–8) c a = 10 c s . b 0 b b b b b b
small concentrations of acid). This is explained by the 2– S O4
fact that a considerable fraction of ions, which form as a result of copper sulfate dissociation, combine –
with H+ ions yielding HS O 4 . The concentration of H+ ions is significantly lower than the concentration of –
HS O 4 ions; therefore, the concentration of acid is determined primarily by the concentration of bisulfate. The negatively charged bisulfate ions migrate from the cathode leading to a decrease in their concentration and, consequently, in the concentration of sulfuric acid (Fig. 3, curve 7). A decrease in the acid concentration raises the buoyancy force; as a result, the limiting cur-
rent density and the mass-transfer coefficient increase (Fig. 4). In excess of supporting electrolyte, incomplete dissociation of copper sulfate and sulfuric acid has only weak effect on the limiting current density (the masstransfer coefficient): the limiting current density changes by not more than 10% when the dissociation constants change by several orders of magnitude (Fig. 4). At the moderate and low concentrations of supporting electrolyte, incomplete dissociation should be taken into consideration, because the error of the limiting current density can exceed 50%. The largest error is observed in the second and third versions of calculations. In the second version, a high concentration of
RUSSIAN JOURNAL OF ELECTROCHEMISTRY
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THE LIMITING CURRENT DENSITY OF COPPER ELECTRODEPOSITION C3–C3b
467
C4–C4b Ic
Id
0 0
–0.5
–0.3
–1.0
–0.6 IId
IIc 0
0
–0.5
–0.4
–1.0
0
2
4
η
–0.8
0
2
4
η
Fig. 2. Contd.
neutral species CuSO4(aq) decreases the migration transfer of copper to the cathode leading to a decrease in the limiting current. In the third version, a high concentration of bisulfate ions, which migrate from the cathode, prevents a significant increase in the concentration of sulfuric acid leading to an increase in the limiting current. For these versions, different dependences of mass-transfer coefficient on the concentration of copper sulfate are observed: an increase in the concentration of copper sulfate leads to a decrease in the masstransfer coefficient for the second version of calculations (Fig. 4, curves 2 and 6) and to an increase in it for the third version of calculation (Fig. 4, curves 4 and 8). In the works of Newman [7, 9], the dependences of limiting current for the complete dissociation and the absence of dissociation of bisulfate ions, corresponding to Fig. 4, curves 3, 7 and 4, 8, were obtained. In these works, the homogeneous chemical reactions were RUSSIAN JOURNAL OF ELECTROCHEMISTRY
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ignored in the mathematical model; therefore, in the absence of dissociation of bisulfate ions, the calculations were performed only at r > 0.5. The mathematical model that accounts for the homogeneous chemical reactions, which is used here, enabled us to calculate the mass-transfer coefficient also at r < 0.5 (Fig. 4, curves 4 and 8), when, along with bisulfate ions, the sulfate ions are present in the solution. It is found that the limiting current (the mass-transfer coefficient), which is obtained under the assumption of virtually complete dissociation of copper sulfate and sulfuric acid by the second stage (version 4) (Fig. 4, curves 4 and 8), agrees well with that derived by taking into account the concentration dependences of equilibrium constants (Fig. 4, curves 1 and 5). This is explained by partial compensation of two effects: a decrease in the limiting current with partial dissociation of copper sulfate and an increase in the limiting current No. 4
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468
VOLGIN, DAVYDOV C5–C5
Ιe
b
V
0
0.4
–0.4
0.2
–0.8
If
0
IIe
IIf
0 0.4
–0.4 0.2
–0.8
0
2
η
4
2
0
4
η
Fig. 2. Contd.
Cac–Cab 0.4
0.2
5 6 7 8
1 2 3 4
0
0
0.4
r
0.8
Fig. 3. Plots of dimensionless concentration of sulfuric acid near the cathode surface C a vs. the relative concentration of sulfuric c
acid, which were calculated: (1 and 5) taking into account the concentration dependences of equilibrium constants Ks and Ka; (2 and 6) ignoring the concentration dependences of equilibrium constants ( c s Ks = K s and c s Ka = K a ); (3 and 7) c s Ks = 106 and b
0
b
0
b
c s Ka = K a ; (4 and 8) c s Ks = c s Ka = 106; (1–4) c s = 10 mol/m3; and (5–8) c s =100 mol/m3. b
0
b
b
b
b
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THE LIMITING CURRENT DENSITY OF COPPER ELECTRODEPOSITION
469
KL
3. Tobias, C.W., Eisenberg, M., and Wilke, C.R., J. Electrochem. Soc., 1952, vol. 99, p. 359.
1.2
4. Wilke, C.R., Eisenberg, M., and Tobias, C.W., J. Electrochem. Soc., 1953, vol. 100, p. 513. 1 2 3 4
5. Maru, Y., Ito, S., Oyama, S., and Kondo, Y., Denki Kugaku, 1970, vol. 38, p. 343.
5 6 7 8
6. Grigin, A.P. and Davydov, A.D., J. Electroanal. Chem., 2000, vol. 493, p. 16.
0.9
7. Newman, J.S., Electrochemical Systems, New York: Prentice-Hall, 1973. 8. Grigin, A.P. and Davydov, A.D., Elektrokhimiya, 1998, vol. 34, p. 1237 [Russ. J. Electrochem. (Engl. Transl.), vol. 34, p. 1111]. 0.6
0
0.4
0.8
r
Fig. 4. Plots of mass-transfer coefficient vs. the relative concentration of sulfuric acid (for notations see Fig. 3).
9. Selman, J.R. and Newman, J., J. Electrochem. Soc., 1971, vol. 118, p. 1070. 10. Volgin, V.M., Grigin, A.P., and Davydov, A.D., Elektrokhimiya, 2003, vol. 39, p. 371 [Russ. J. Electrochem. (Engl. Transl.), vol. 39, p. 335].
with partial dissociation of sulfuric acid by the second stage.
11. Langmuir, D.L., Aqueous Environmental Geochemistry. Upper Saddle River, New Jersey: Prentice Hall, 1997.
CONCLUSIONS
12. Pitzer, K.S., Activity Coefficients in Electrolyte Solutions, Boca Raton, FL: Florida: CRC Press, 1991.
The limiting current of copper electrodeposition on vertical plane electrode under the conditions of natural convection of copper sulfate and sulfuric acid solution is determined using the numerical solution of the problem taking into account the homogeneous chemical reactions and the concentration dependences of physicochemical properties of solution. A comparison of the results, which were obtained taking into consideration and ignoring the concentration dependences of dissociation constants of copper sulfate and sulfuric acid by the second stage, and also the data for the limiting cases of virtually complete dissociation of copper sulfate and sulfuric acid showed that the limiting current (the masstransfer coefficient), which is obtained under the assumption of virtually complete dissociation of copper sulfate and sulfuric acid by the second stage, agrees well with that derived by taking into account the concentration dependences of equilibrium constants. Thus, for calculating the limiting current density of copper electrodeposition under the natural convection, it is necessary to take into consideration the homogeneous chemical reactions and the concentration dependences of dissociation constants or to assume (for simplicity) that the complete dissociation of copper sulfate and sulfuric acid takes place.
13. Casas, J.M., Alvarez, F., and Cifuentes, L., Chem. Eng. Sci., 2000, vol. 55, p. 6223. 14. Samson, E., Marchand, J., and Beaudoin, J.J., Cem. Concr. Res., 2000, vol. 30, p. 1895. 15. Samson, E., Lemaire, G., Marchand, J., and Beaudoin, J.J., Comput. Mater. Sci., 1999, vol. 15, p. 285. 16. Awakura, Y., Doi, T., and Majima, H., Metall. Trans. B, 1988, vol. 19B, p. 5. 17. Baes, C.F., Reardon, E.J., and Moyer, B.A., J. Phys. Chem., 1993, vol. 97, p. 12343. 18. Wu, B.-H., Wan, C.-C., and Wang, Y.-Y., J. Electrochem. Soc., 2003, vol. 150, p. C7. 19. Levich, V.G., Physicochemical Hydrodynamics, Englewood Cliffs (NJ): Prentice-Hall, 1962. 20. Gebhart, B., Jaluria, I., Mahajan, R.I., and Sammakia, B., Buoyancy-Induced Flows and Transport New York: Hemisphere, 1988. 21. Samarskii, A.A. and Gulin, A.V., Chislennye metody (Numerical Methods), Moscow: Nauka, 1989. 22. Spravochnik po elektrokhimii (Handbook of Electrochemistry), Sukhotin, A.M., Ed., Leningrad: Khimiya, 1981.
REFERENCES
23. Spravochnik khimika (Handbook of Chemists), Nikol’skii, B.P., Ed., vol. 3, Moscow: Khimiya, 1965.
1. Levich, V.G., Zh. Fiz. Khim., 1948, vol. 22, p. 575. 2. Wagner, C., Trans. Electrochem. Soc., 1949, vol. 95, p. 161.
24. Duchanoy, C., Lapicque, F., Oduoza, C.F., and Wragg, A.A., Electrochim. Acta, 2000, vol. 46, p. 433.
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