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The meaning of exchange current densities in electrocrystallization
423
J. Electroad Chem., 229 (1987) 423-427 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
THE MEANING OF EXCHANGE IN ELECTROCRYSTALLIZATION
CURRENT
DENSITIES
l
E.B. BUDEVSKI Central Laboratory of Electrochemical
Power Sources, Bdgarian
Academy of Sciences, Sofa 1040 (Bulgaria)
(Received 20th February 1987)
INTRODUCTION
The electrochemical deposition of metals proceeds in general on highly non-homogeneous surfaces. This is also true even in the case of deposition on single-crystal faces. The kinetic equations of the process of deposition include parameters which depend on the energy profile of the path of transfer, such as rate constants or the related exchange current densities, transfer coefficients, concentrations of intermediates - like adatoms, etc. Several transfer paths can be distinguished in the process of deposition. Until recently the surface diffusion path was considered as the energetically most favourable one. It has been found, however, that in some cases direct transfer to sites of growth of the crystal face is predominant [l]. This fact shows only that an estimation of the prevailing mechanism based on theoretical considerations can be misleading, and if direct transfer has a significant role the deposition is not uniformly distributed over the surface. The treatment of the deposition process, taking into account that, for the deposition and dissolution, not all sites on the crystal surface are equivalent, can be made by using the attachment/detachment frequencies wdep,, and wdiss., to and from a site i, introduced by Kaishew as early as 1945 [2]: Wdep,r
=p,c
wdiss,i
=
q1
exp( -&e,,,/W exp( - AG,,JkT)
exd
-B%W~~)
(1)
exp[ (1 - /?)ze,E/kT]
(2)
where AGdep, and AGdiss,,are the activation energies for the deposition/dissolution process for site i at E = 0,q and p are the rate constants for the cathodic and the anodic processes, respectively and c is the concentration of depositing ions in the solution.
* In honour of Professor H. Gerischer on the occasion of his retirement as Director of the Fritz-Haber Institute.
Fig. 1. The structure of a crystalline singular face and sites of exchange: “ad”, adatom; ” P~acancy; “k”, half-crystal (kink) site; ad-ex, exchange with adatoms; k-ex, exchange with kink atoms.
On a close-packed, equ~b~um form (or singular) face, different sites for the deposition or the dissolution processes can be distinguished, as shown in Fig. 1. The most rudimentary of them are the adatom sites “ad”, the free surface sites “f”, the vacancy sites “v” and the kink sites “k” (or the sites of half-crystal position as introduced by Stranski and Kaischew). Some of the sites are, as seen, conjugated. For instance, a free surface site is transformed into an adatom site by deposition and, inversely, an adatom site is transformed into a free surface site by dissolution, or in general terms, dep $,i “- “i + 1” d:s
This reaction equation means that by deposition of an atom to a site “i ” this site is transformed into its conjugated one “i + 1” and vice versa by dissolution. Each act of attachment or detachment results in a change in the number of sites “i ” and their conjugated sites “i + 1”. The only site that is exempt from this rule is the kink site. As can be easily seen, if an atom is attached to or detached from a kink site the kink site position is actually restored, regardless of the displa~me~t by one atom position. It is essential to underline that no change of the surface sites results by such an act dep “‘k“*“k” diss
’
If Q (in cm2) is the area occupied by one atom on the surface of a singular crystal face, then l/51 = N would be the number of sites per unit area available for an
425
attachment/detachment process. The area ratio occupied by all sites of type ‘3” is n/N, where n, is the number of sites of type “i” per cm*. It is evident that all the sites of type “i ” contribute to the overall exchange rate proportional to the part of the area occupied by them and their exchange frequencies wdW,, and Wdiss,r. ADATOMS -
EXCHANGE CURRENT DENSITY AND CONCENTRATION
The adatom can be regarded as an intermediate species on the reaction path of a solution ion to an atom belonging to the crystal lattice. The solution ion is transformed by discharge intermediately to an atom adsorbed on the surface and is then incorporated into the crystal lattice on a kink site by surface displacement. As long as the exchange between adatoms and kink atoms is uninhibited and sufficiently fast, the adatom concentration is defined, on the one hand, by the energy IV of the kink-adatom transfer: Ftad = n, exp( - W/kT)
= n,91
(3)
The pre-exponential term includes an entropy factor and is roughly equal to the number nt of sites available for adsorption. As long as no potential gradients exist parallel to the metal surface, the energy W should not depend on the electrode potential and the term n, exp( - W/kT) must be a constant defining the equilibrium value of the adatom concentration n,“d. On the other hand, the exchange rate of adatoms with ions in the solution is given by the exchange frequencies wd_,f and w~_~. The cathodic current density by which adatoms are formed is determined by the attachment frequency wdep,r to free surface sites “ft: lC.,d= 4monwdep,fnf/N
(4)
Because of the usually low concentration of adatoms the last term in eqn. (4), n f/N, is close to unity. The anodic current density resulting from dissolution of adatoms is l,,ad
=
4 n-eonw
d&,&d n adjN
(51
At the equilibrium potential, E = E,, the cathodic and anodic current densities are equal and can be defined as the exchange current density of the adatoms: had = bd = ‘a,ad. Then from eqns. (4) and (5) n ad/N
=
wdep.f/w
diss,ad
(6)
Having in mind eqris. (1) and (2) and introducing the new constant K = f Pf/%d)
exP@%iss,ad -
Af;dep,t )
we obtain nad = NKc exp( - ze,E,/kT)
= n,”
(7) is obviously equal to the desorption energy of The term (A%issSad- k&,,t) adatoms at E = 0. Because of the c-E, Nemst relation, nad is a constant = nzd independent of the value of E, or c separately.
426
Let us now consider a crystal face on which the exchange between adatoms and kink atoms is blocked. This case can occur either by inhibition of the exchange rate or by reducing the number of kink sites, as, for instance, in the case of an intact dislocation-free face. An increase of the electrode potential, say to the negative side, would then result in an increase of the cathodic fhuc and a gradual increase of the adatom concentration. The anodic flux, which at equilibrium makes the balance with the cathodic fhtx, depends, as seen from eqns. (5) and (2), not only on the potential but also on the adatom concentration. Therefore, after the initial decrease following the negative potential change, the anodic flux begins to rise as the adatom concentration increases. A balance with the cathodic flux is reached at a new, higher adatom concentration derived from eqn. (6), which holds again because of the equality of the cathodic and anodic fhtxes. The derived equation is similar to eqn. (7) but contains the imposed non-equilibrium potential E instead of the equilibrium potential E,. Dividing eqn. (7) by this new equation we obtain hd=
nz
exp( -ze&kT)
(8)
where 17= E, - E is the overpotential (cathodic taken as positive). The respective exchange current density of the adatoms can be derived from eqn. (4) or (5): iO,ad=honPfc
exp(-beOEr/kT)
(9)
The exchange current density of the adatoms is a well-defined quantity depending on the ion concentration c in the solution only, which for its part defines the value of the equilibrium potential. THE KINK SITE
As already mentioned, this position has singular features. The act of deposition or dissolution to and from this site results in reproduction of the site. For the cathodic and anodic currents of these sites we can write P-0
Ic,k= qmon wd,,knkiN
and iak= , %mmwdiss,knk/N
01)
Both currents must equalize at equilibrium, i.e. at the equilibrium potential E,. It follows immediately from eqns. (10) and (11) that the attachment/detachment frequencies must be equal: wdep,k(Er) = wbt(Er), and that a kink site is half the time occupied and the other half free. This fact is of importance for the definition of the equilibrium conditions. With eqns. (1) and (2), it follows from this equality that ze,E, = kT ln( pJ&)
+ AGk -t- kT ln c
(12)
where AGk = AG,, k - AGd_,k is the Gibbs energy of the transfer of an atom from a kink site to a solution ion at E = 0. Equation (12) is another form of the Nemst equation.
427
We can also define the exchange current density of the kink atoms: i,, = i,, = iak or i,, = con&.( n&V)c = const.(nJN)
exp( - ~~e~E~/kT) exp[(l
- @)ze,E,/kTj
(13)
The constant includes constants such as q_,, p, q and the exponential AG term. There are two important conclusions that can be made based on these considerations: (i) The exchange current density iqL of the kink atoms is not a constant and depends on the number of kink sites nk, or, in other words, on the step density if steps are present on the crystal surface. As long as the step density is undefined and depends on the growth or preparation conditions of the surface, the exchange current density is an undefined quantity. (ii) This is also true for the overall exchange current density, composed of the exchange current densities of the sites of all types, when the exchange cnrrent density of the kink sites makes a significant contribution. In my opinion, this explains the large deviation of the values of the exchange current densities reported for metal surfaces. Without going into further detail, I would like to point out only that relations for the current of growth, i.e. for the deposition process, can be readily obtained using the attac~ent/detac~ent frequencies introduced by KaiSchew (see, for example, ref. 3).
CONCLUSIONS
(a) The ~~~b~urn adatom concentration n,91is a constant characteristic for the metal-metal ion system independent of the ion concentration. (b) The adatom concentration may be changed by changing the potential in the overpotential region if the exchange with kink atoms is blocked or slow. (c) The exchange current density of adatoms is a constant for the system but depends on the ion concentration. (d) The exchange current density of kink atoms is not a constant and depends on the kink site density, i.e. on the step density or, in other words, on the preparation conditions.
REFERENCES 1 T. Vitanov, A. Popov and E. Budevski, J. EIectrochem. Sot., 121 (1974) 207. 2 R. Kaischew, Annu. Univ. Sofia Fat. Physiwmathem. (Chim.), 42 (1945/1946) 109. 3 E. Budevski in B.E. Conway, .V.O’M. Bock&, E. Yeager. S.U.M. Khan and R.E. White (Eds.), Comprehensive Treatise of Electrochemistry, Vol. 7, Plenum Press, New York. 1983, Ch. 7, p. 399.
J. Electroad Chem., 229 (1987) 423-427 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
THE MEANING OF EXCHANGE IN ELECTROCRYSTALLIZATION
CURRENT
DENSITIES
l
E.B. BUDEVSKI Central Laboratory of Electrochemical
Power Sources, Bdgarian
Academy of Sciences, Sofa 1040 (Bulgaria)
(Received 20th February 1987)
INTRODUCTION
The electrochemical deposition of metals proceeds in general on highly non-homogeneous surfaces. This is also true even in the case of deposition on single-crystal faces. The kinetic equations of the process of deposition include parameters which depend on the energy profile of the path of transfer, such as rate constants or the related exchange current densities, transfer coefficients, concentrations of intermediates - like adatoms, etc. Several transfer paths can be distinguished in the process of deposition. Until recently the surface diffusion path was considered as the energetically most favourable one. It has been found, however, that in some cases direct transfer to sites of growth of the crystal face is predominant [l]. This fact shows only that an estimation of the prevailing mechanism based on theoretical considerations can be misleading, and if direct transfer has a significant role the deposition is not uniformly distributed over the surface. The treatment of the deposition process, taking into account that, for the deposition and dissolution, not all sites on the crystal surface are equivalent, can be made by using the attachment/detachment frequencies wdep,, and wdiss., to and from a site i, introduced by Kaishew as early as 1945 [2]: Wdep,r
=p,c
wdiss,i
=
q1
exp( -&e,,,/W exp( - AG,,JkT)
exd
-B%W~~)
(1)
exp[ (1 - /?)ze,E/kT]
(2)
where AGdep, and AGdiss,,are the activation energies for the deposition/dissolution process for site i at E = 0,q and p are the rate constants for the cathodic and the anodic processes, respectively and c is the concentration of depositing ions in the solution.
* In honour of Professor H. Gerischer on the occasion of his retirement as Director of the Fritz-Haber Institute.
Fig. 1. The structure of a crystalline singular face and sites of exchange: “ad”, adatom; ” P~acancy; “k”, half-crystal (kink) site; ad-ex, exchange with adatoms; k-ex, exchange with kink atoms.
On a close-packed, equ~b~um form (or singular) face, different sites for the deposition or the dissolution processes can be distinguished, as shown in Fig. 1. The most rudimentary of them are the adatom sites “ad”, the free surface sites “f”, the vacancy sites “v” and the kink sites “k” (or the sites of half-crystal position as introduced by Stranski and Kaischew). Some of the sites are, as seen, conjugated. For instance, a free surface site is transformed into an adatom site by deposition and, inversely, an adatom site is transformed into a free surface site by dissolution, or in general terms, dep $,i “- “i + 1” d:s
This reaction equation means that by deposition of an atom to a site “i ” this site is transformed into its conjugated one “i + 1” and vice versa by dissolution. Each act of attachment or detachment results in a change in the number of sites “i ” and their conjugated sites “i + 1”. The only site that is exempt from this rule is the kink site. As can be easily seen, if an atom is attached to or detached from a kink site the kink site position is actually restored, regardless of the displa~me~t by one atom position. It is essential to underline that no change of the surface sites results by such an act dep “‘k“*“k” diss
’
If Q (in cm2) is the area occupied by one atom on the surface of a singular crystal face, then l/51 = N would be the number of sites per unit area available for an
425
attachment/detachment process. The area ratio occupied by all sites of type ‘3” is n/N, where n, is the number of sites of type “i” per cm*. It is evident that all the sites of type “i ” contribute to the overall exchange rate proportional to the part of the area occupied by them and their exchange frequencies wdW,, and Wdiss,r. ADATOMS -
EXCHANGE CURRENT DENSITY AND CONCENTRATION
The adatom can be regarded as an intermediate species on the reaction path of a solution ion to an atom belonging to the crystal lattice. The solution ion is transformed by discharge intermediately to an atom adsorbed on the surface and is then incorporated into the crystal lattice on a kink site by surface displacement. As long as the exchange between adatoms and kink atoms is uninhibited and sufficiently fast, the adatom concentration is defined, on the one hand, by the energy IV of the kink-adatom transfer: Ftad = n, exp( - W/kT)
= n,91
(3)
The pre-exponential term includes an entropy factor and is roughly equal to the number nt of sites available for adsorption. As long as no potential gradients exist parallel to the metal surface, the energy W should not depend on the electrode potential and the term n, exp( - W/kT) must be a constant defining the equilibrium value of the adatom concentration n,“d. On the other hand, the exchange rate of adatoms with ions in the solution is given by the exchange frequencies wd_,f and w~_~. The cathodic current density by which adatoms are formed is determined by the attachment frequency wdep,r to free surface sites “ft: lC.,d= 4monwdep,fnf/N
(4)
Because of the usually low concentration of adatoms the last term in eqn. (4), n f/N, is close to unity. The anodic current density resulting from dissolution of adatoms is l,,ad
=
4 n-eonw
d&,&d n adjN
(51
At the equilibrium potential, E = E,, the cathodic and anodic current densities are equal and can be defined as the exchange current density of the adatoms: had = bd = ‘a,ad. Then from eqns. (4) and (5) n ad/N
=
wdep.f/w
diss,ad
(6)
Having in mind eqris. (1) and (2) and introducing the new constant K = f Pf/%d)
exP@%iss,ad -
Af;dep,t )
we obtain nad = NKc exp( - ze,E,/kT)
= n,”
(7) is obviously equal to the desorption energy of The term (A%issSad- k&,,t) adatoms at E = 0. Because of the c-E, Nemst relation, nad is a constant = nzd independent of the value of E, or c separately.
426
Let us now consider a crystal face on which the exchange between adatoms and kink atoms is blocked. This case can occur either by inhibition of the exchange rate or by reducing the number of kink sites, as, for instance, in the case of an intact dislocation-free face. An increase of the electrode potential, say to the negative side, would then result in an increase of the cathodic fhuc and a gradual increase of the adatom concentration. The anodic flux, which at equilibrium makes the balance with the cathodic fhtx, depends, as seen from eqns. (5) and (2), not only on the potential but also on the adatom concentration. Therefore, after the initial decrease following the negative potential change, the anodic flux begins to rise as the adatom concentration increases. A balance with the cathodic flux is reached at a new, higher adatom concentration derived from eqn. (6), which holds again because of the equality of the cathodic and anodic fhtxes. The derived equation is similar to eqn. (7) but contains the imposed non-equilibrium potential E instead of the equilibrium potential E,. Dividing eqn. (7) by this new equation we obtain hd=
nz
exp( -ze&kT)
(8)
where 17= E, - E is the overpotential (cathodic taken as positive). The respective exchange current density of the adatoms can be derived from eqn. (4) or (5): iO,ad=honPfc
exp(-beOEr/kT)
(9)
The exchange current density of the adatoms is a well-defined quantity depending on the ion concentration c in the solution only, which for its part defines the value of the equilibrium potential. THE KINK SITE
As already mentioned, this position has singular features. The act of deposition or dissolution to and from this site results in reproduction of the site. For the cathodic and anodic currents of these sites we can write P-0
Ic,k= qmon wd,,knkiN
and iak= , %mmwdiss,knk/N
01)
Both currents must equalize at equilibrium, i.e. at the equilibrium potential E,. It follows immediately from eqns. (10) and (11) that the attachment/detachment frequencies must be equal: wdep,k(Er) = wbt(Er), and that a kink site is half the time occupied and the other half free. This fact is of importance for the definition of the equilibrium conditions. With eqns. (1) and (2), it follows from this equality that ze,E, = kT ln( pJ&)
+ AGk -t- kT ln c
(12)
where AGk = AG,, k - AGd_,k is the Gibbs energy of the transfer of an atom from a kink site to a solution ion at E = 0. Equation (12) is another form of the Nemst equation.
427
We can also define the exchange current density of the kink atoms: i,, = i,, = iak or i,, = con&.( n&V)c = const.(nJN)
exp( - ~~e~E~/kT) exp[(l
- @)ze,E,/kTj
(13)
The constant includes constants such as q_,, p, q and the exponential AG term. There are two important conclusions that can be made based on these considerations: (i) The exchange current density iqL of the kink atoms is not a constant and depends on the number of kink sites nk, or, in other words, on the step density if steps are present on the crystal surface. As long as the step density is undefined and depends on the growth or preparation conditions of the surface, the exchange current density is an undefined quantity. (ii) This is also true for the overall exchange current density, composed of the exchange current densities of the sites of all types, when the exchange cnrrent density of the kink sites makes a significant contribution. In my opinion, this explains the large deviation of the values of the exchange current densities reported for metal surfaces. Without going into further detail, I would like to point out only that relations for the current of growth, i.e. for the deposition process, can be readily obtained using the attac~ent/detac~ent frequencies introduced by KaiSchew (see, for example, ref. 3).
CONCLUSIONS
(a) The ~~~b~urn adatom concentration n,91is a constant characteristic for the metal-metal ion system independent of the ion concentration. (b) The adatom concentration may be changed by changing the potential in the overpotential region if the exchange with kink atoms is blocked or slow. (c) The exchange current density of adatoms is a constant for the system but depends on the ion concentration. (d) The exchange current density of kink atoms is not a constant and depends on the kink site density, i.e. on the step density or, in other words, on the preparation conditions.
REFERENCES 1 T. Vitanov, A. Popov and E. Budevski, J. EIectrochem. Sot., 121 (1974) 207. 2 R. Kaischew, Annu. Univ. Sofia Fat. Physiwmathem. (Chim.), 42 (1945/1946) 109. 3 E. Budevski in B.E. Conway, .V.O’M. Bock&, E. Yeager. S.U.M. Khan and R.E. White (Eds.), Comprehensive Treatise of Electrochemistry, Vol. 7, Plenum Press, New York. 1983, Ch. 7, p. 399.