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Chronoamperometry for the determination of metallic interdiffusion coefficients. Rapid transport processes in the first atomic layers
JOURNAL OF
ELSEVIER
Journal of Eiectroanalytical Chemistry 396 (1995) 203-209
Chronoamperometry for the determination of metallic interdiffusion coefficients. Rapid transport processes in the first atomic layers l Fr6d6ric Lantelme, Marius Chemla Laboratoire d'Electrochimie, UniversiM Pierre et Marie Curie, 4 Place Jussieu, F 75252 Paris Cedex 05, France Received 17 January 1995; in revised form 3 February 1995
Abstract
The potential step technique was used to study the interdiffusion in the solid state at high temperature using a molten electrolyte. The method was applied to several binary metal systems and proved to be efficient for the determination of the interdiffusion coefficient D as a function of the temperature and composition of the generated alloys. The electrochemical response at short times showed a net departure from the usual diffusion laws and could be interpreted in terms of a very fast transport in the initial layers of the metal interface. Analysis of the transients was carded out by digital simulation in order to take into account the dependence of D on the concentration and distance. Keywords: Chronoamperometry; lnterdiffusion coefficient; Binary metal system
1. Introduction Electrodeposition from fused electrolytes has been used to form protective metallic coatings with good anticorrosion properties. Three electrolytic deposition processes have been developed: (a) electroplating, which produces a constant composition layer of metal coating on the substrate; (b) metalliding, which involves deposition and solid state intermetallic diffusion; (c) alloy deposition by simultaneous electroreduction of two or more cations. When the deposition is carried out at high temperature, the kinetics of formation of an intermetallic diffusion layer are fast and result in the growing of an intermediate alloy and good adhesion of the deposited coating to the substrate metal [ 1]. These technical applications of electrocoating from molten salts involve the study of electrochemical reactions which proceed under the control of the diffusion of electroactive species: diffusion of ions in the liquid electrolyte and of metallided atoms into the solid substrate. These two kinds of diffusion in the electrolyte and in the solid state are also involved when the coating is built up by the codeposition of several elements; the composition of the coating depends on the diffusion fluxes in the electrolyte and on the interdiffusion in the coating [2].
i Dedicated to Professors K. Honda, H. Matsuda and R. Tamamushi on the occasion of their 70th birthdays and in recognition of their contribution to electrochemistry. 0022-0728/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 0 2 2 - 0 7 2 8 ( 9 5 ) 0 3 9 3 3 - 3
These mechanisms can be studied quantitatively by the use of transient electrochemical techniques. The response is dependent on the time scale under consideration. Most generally, as a consequence of the simple electrochemical reactions and the high temperature, the electron transfer is fast enough to give a reversible response; then the surface activity of the deposited metal is controlled by the electrode potential. Detailed examination of the experimental results reveals that additional effects should be taken into account in order to perform a correct analysis of the transient responses. (1) There is a shift in the electrode lelectrolyte interface during the passage of the current due to metal deposition or dissolution. The influence of this effect has already been studied and mathematical analyses have been performed [3,4]. (2) The interdiffusion coefficient [5] depends on the alloy composition, which introduces a large perturbation in the analysis of transients. In a previous paper [6], we have shown that this influence does not modify the classical Cottrell relation, i.e. the transient current density j remains proportional to the reciprocal of the square root of the time t, j = A/v/-[. However, the dependence of the proportionality factor A on the diffusion coefficient and on the concentration has not yet been established. (3) At very short times, the response contains additional
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F. Lantelmeand M. Chemla/ Journalof ElectroanalyticalChemistry396 (1995)203-209
terms, such as double-layer capacity charging or electroreduction of adsorbed species. From our present study, we observed that in most experiments the current density is enhanced with respect to the predicted values at very short times. This result could be assigned to a rapid motion of the deposited atoms in the initial layers of the interface. This article is devoted to the analysis of the transient response, which includes the influence of these various phenomena that are not included in the conventional treatment. To obtain a better understanding of this behaviour, we present first a description of the experimental procedure. These transport problems cannot be solved algebraically, and therefore in order to obtain fundamental information, the electrochemical results are analysed using the digital simulation method [7].
2. Experimental The method involves the use of polarization cells which have the general configuration Alloy electrode Liquid electrolyte I Pure M M-M' containing Mn + ,
or metal atoms M) in the bulk of the two phases (electrolyte and metal) obey the equation
OCj( x, t) 0 I Cj( x, t) l at = 7x Dj o-----T--
(2)
where Dj and cj are the diffusion coefficient and concentration of the electroactive species respectively, and x is the distance normal to the electrode. The initial boundary condition is
q ( x, 0) = c 3"
(3)
c[ is the initial concentration of the electroactive species. The thickness of the diffusing medium is considered to be large compared with the diffusion layer; the condition of semi-infinite diffusion is then fulfilled cj(o¢, t) = cj"
(4)
The flow of metal from the surface into the bulk of this electrode by diffusion is limited by the flux of metal J formed by electrodeposition and equal to it. Therefore the current density j associated with the perturbation is (j being positive for an oxidation process) ~x
j = - nFJ = - nFD i
x= 0
(5)
(1)
M is the electroactive metal and M' is the metal substrate. In most of our experiments, the electrolyte was a eutectic LiC1 + KCI mixture whose component salts were obtained from Merck (analytical reagen0. Before each experiment, the mixture was heated in the cell at 130°C under vacuum for 4 h, and then heated at the experiment temperature (400-540°C) while bubbling HCI for 6 b. Finally, argon gas was circulated for a few hours to eliminate HCI. The metal ions M were generated by the anodic dissolution of a rod of pure M; the counterelectrode was a chlorine electrode [8] placed in a separate compartment connected with a porous tube. The chlorine electrode was also used as a reference electrode. For the measurements using cell (1), the secondary reference electrode and the counterelectrode were made of a rod of pure M. The working electrode was a wire, I mm in diameter, of Au-Cu alloy. An electronic device built in our laboratory was used to deliver potentiostatic pulses. The potential pulses and current responses were recorded on a double-trace digital oscilloscope Nicolet 310. The purity of the molten electrolyte was examined by cyclic voltammetry. The residual current density measured with a platinum electrode over the potential range -0.5, - 1 . 8 V vs. CI 2 IC1- was less than 0.1 m A c m -2.
which is available for both diffusion of M n+ in the liquid phase and of M atoms in the solid metal substrate (Fig. 1). The concentration profile c j ( x , t) is calculated by integrating the diffusion equation (Eq. (2)) with boundary conditions expressing the electrochemical constraints arising from the experimental conditions. For example, using the potentiostatic method, the variation of the current density vs. time should obey the Cottrell law, provided that the diffusion constant D is a constant, following the expression j = n F ( cj - cj" ) ~ j /
(6)
Trt
This expression is valid for the determination of Dj in the liquid phase and in the solid metal [6]. In fact, the
etal• c
CsM
Solid m
C"M
Liquid
phase
C'Mn+
CSMn+
Distance
3. Basic principle of the method When planar diffusion is the rate-determining step, the concentration changes of electroactive species j (ions, M "+
x
Fig. 1. Profile of the concentrations of electroactive species during electrochemical reduction in both the electrolyte and the solid substrate phase.
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F. Lantelme and M. Chemla/ Journal of Electroanalytical Chemistry 396 (1995) 203-209
concentration changes in the metal and the electrolyte are interrelated at any time by CM -- CM / DM"+ "+ -- CM,+ * = V OM cM
(7)
The diffusion coefficient in the solid metal is generally smaller by a factor of 10 6 compared with the melt, and the potential step is easily adjusted to appropriate values to make the current only dependent on the solid state interdiffusion; the difference CM,+-- Cry,+ is very small, and thus the values of c M and c M are determined from the Nernst law by the values of the imposed potentials provided that the activity coefficients of the components are known. In most cases, thermodynamic data are available in the literature and may be expressed by a Margules [9] expansion In T - a 2 / 2 ( 1 - X ) z + a 3 / 3 ( 1 - - X ) 3 + a4/4(1
--X)4 -[- ...
(8)
where X is the molar fraction of species j. The potential step can be chosen to be either positive or negative with respect to the equilibrium value in order to measure Dj when diffusion takes place in or out of the sample material. Most generally, for a given composition, we observed that the experimental Dj values do not depend on whether the diffusion takes place in or out of the substrate. Moreover, the variation of the amplitude of the potential perturbation permits surface alloys to be generated with different compositions. The method therefore provides an efficient tool for the determination of D M values as a function of concentration. Three mathematical difficulties appear in the solution of the diffusion equation (Eq. (2)) corresponding to a given technique. (1) The diffusion coefficient of the deposited metal may be quite different from that of the substrate. This property induces a Kirkendall effect and leads to a movement of the internal reference frame. Only an interdiffusion coefficient Dj is then observed. (2) The interdiffusion coefficient of the metal is strongly dependent on the composition of the alloy; thus a variable Dj should be introduced along the concentration gradient. (3) The deposition of the metal results in a shift of the interface which must be accounted for as a moving origin of the abscissae. For these conditions, the conventional equations of chronoamperometry and chronopotentiometry are not applicable. New mathematical expressions have been developed to account for the moving boundary [4]. However, when perturbing phenomena, such as the concentration and distance dependence of Dj, are superimposed on the condition of the moving boundary, the problem is not solvable, Digital simulation is then the only way to treat the phenomena comprehensively. The outline of the implementation of this technique is described below.
It is assumed that the diffusion coefficient Dj depends on both the concentration cj of the active species and the distance x to the interface. The rate of movement of the interface (9)
v =j/nFVj
is also taken into account [4]; Vj is the partial molar volume of the electroactive species. For a metallic solid solution, the sign of the current density is positive when the electroactive metal is dissolved (oxidation), i.e. when the interface moves towards the bulk of the sample. Then the diffusion equation (Eq. (2)) is written as 0cj( x, t) ~t
=
aDj 0cj ~x ~x
- - - -
+
02cJ OJ - ~ x2
+
v ~cj -0x
(lO)
Dj depends on x and cj which is also dependent on x; thus d Dj
ODj --
dx
_
_
~x
aDj Ocj -[-
-
-
0cj ax
Eq. (10) is solved by the technique of numerical simulation using the point method [7]. The time step At is a fraction of the total duration t T of the experiment (At ~/T/10000). According to the usual convergence requirements [7], the space interval was chosen to be equal to: /ix = 2.1~LZ1"~; D L is the largest diffusion coefficient involved in the experiment. Naturally, the convergence of the technique was carefully checked using mesh refinement methods. We now examine separately the two cases where Dj depends either on the concentration or on the distance to the interface, the shift of the interface being neglected. The interpretation of experiments where Dj depends on both the concentration and distance will be examined later. 3.1. Dependence o f D on concentration
Two cases are considered: a linear dependence Dj = Dj°(1 + hcj)
(lla)
and an exponential dependence Oj = Oi ° exp(hcj)
(1 lb)
where Dj° is the limiting value of the diffusion coefficient at zero c and h is a constant parameter. The numerical treatment of Eq. (10) indicates that the current always remains proportional to the reciprocal of the square root of time as in the Cottrell law. This property of the potentiostatic method has been previously demonstrated using the Boltzmann transformation [6], since the limiting conditions of the method can be expressed with a non-dimensional variable x/D~'fl. The linear relationship of j vs. I / ~ is valid even if Dj is dependent on c and regardless of the variation of Dj as a function of cj [6]. The slope of the linear variation of j vs. the reciprocal of the square root of t was used to determine a particular value of the diffusion
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F. Lantelme and M. Chemla/ Journal of Electroanalytical Chemistry396 (1995) 203-209
coefficient, D ; °tt, simply by applying the Cottrell equation (Eq. (6)). The numerical treatment indicated that D; °tt is a value of Dj deduced from Eq. (lla) or (lib) for a concentration, C; °tt, n e a r the interface. Assuming that the variations of the diffusion coefficient are not too large, this concentration was computed to be approximately equal to C; °tt~" 0 . 6 4
c~ + 0.36cj*
:C......Z' C'- C' 0,8
0.6
(12) 0.4
where c~ is the surface concentration and cj* is the bulk concentration.
.C1~I
_ - -
0.2
3.2. Dependence of D on distance 2
This dependence has been studied recently to take into account the results of transient techniques when the observation time is short enough to detect the transport phenomena in the initial atomic layers near the interface [2]. It has been proposed that the increase in transport kinetics in this region can be represented by introducing a variable diffusion coefficient which increases as the distance to the interface decreases. Here, as in our previous article [2], it was assumed that Dj obeys the equation Dj
=
Dj* e x p ( 8 / x )
(13)
where D[ is the bulk interdiffusion coefficient and 8 is a characteristic distance called the relevant rapid diffusion thickness (RRDT) [2]. By introducing Eq. (13) into Eq. (2), the diffusion equation to be solved is 0cj(x, t) 8 0Cj 02Cj at = Dj'~-~""~" X +Dj ax 2
(14)
At short distances, Dj becomes very large; thus, in this region, the concentration profile is nearly flat. For the numerical calculation, it was considered that, at a distance shorter than 8' = p8 (with p = 0.25), the concentration c remains constant. The numerical computations show that a Cottrell-type relationship is obeyed only at long times. Departures appear at short times: the ratio between the calculated current density j and the current density Jc deduced from the Cottrell law increases as the time t decreases. The time
I/J¢ 1.5
1.4
1.3
,
,
,
,
0.5 •
.
.
.
.
.
1;o
.
.
1.,~/~ .
Fig. 2. Departure from the Cottrell law, J/Jc vs. l / ~ / ~ ' s ) , j is the calculated current density taking account of the rapid diffusion in a layer of thickness d (RRDT). j~ is the current density deduced from the Cottrell law. Characteristic time of the system, ~'c = ~$2 / 2 Dj °.
,
x/6
Fig. 3. Concentration profile near the alloy surface: (a) calculated from the RRDT concept introducing a variation of the diffusion coefficient near the interface (Eq. (13)); (b) deduced from the classical treatment for
a constantdiffusioncoefficient Dj°. The mean concentration CRSATis calculated for a thickness(RSAT)equal to 8 (Eq. (1 la)). scale at which this departure becomes significant is characterized by the time ~s
rs = 82/2Dj
(15)
The variations in the ratio J/Jc vs. 1 / t ~ ) obtained by numerical computation are reported in Fig. 2. The Cottrell law is only obeyed at large t; for t = 100r~, the departure is about 8%. The RRDT concept has an important consequence for the distribution of electroactive species within the initial surface layers. Indeed, when x becomes small, we observe from the numerical treatment that the slope of the concentration profile tends towards zero, even for small t, in contrast with the value (cj* - c~/~V/-~--Djt, predicted by the classical theory. As indicated in Fig. 3 (curve a), the profile remains very flat near the interface, whereas the conventional calculation indicates a steep concentration change in this region (curve b).
4. Interdiffusion in the copper-gold system Using the basic principles developed above, we examine the results of experiments carried out by the potential step method to study the interdiffusion of copper and gold using cell (1) where M = Cu and M' = Au [6]. The electrolyte was a fused eutectic LiC1 + KC1. The copper ions Cu ÷ were introduced into the bath by the anodic dissolution of a pure copper rod (Johnson-Matthey). The transient currents l(t) observed at a scale of 10 -2 s, after a steep change of the potential, are shown in Fig. 4. The plots in Fig. 4 are in agreement with the theoretical result of a linear dependence of 1 vs. 1/ v~" even when Dj is dependent on c. However, as computed by our numerical treatment, the diffusion coefficients deduced from the Cottrell equation (Eq. (6)) showed a strong dependence on the overpotential r/. This effect corresponds to the change
F. Lantelme and M. Chemla / Journal of Electroanalytical Chemistry. 396 (1995) 203-209 limA
•"
I/~A
207
J ,40 .20
5
• 20 -40 -60
20
Fig. 4. Analysis of chronoamperograms after potential steps for a copper-gold alloy (Eq. (4)). Temperature, 402°C; c~u+ =91 × 10 -3 M in LiC1 + KCI; alloy composition c~u = 13.7 X i 0 - 3 tool c m - 3; electrode area, 1.I 1 cm 2. Values of the potential step: a, a', 200 mV; b, b', 100 mV; c, c', 50 mV; d, d', 20 mV; e, e', 10 mV.
4
0
,t"It21 S.
Fig. 5. Analysis of c~onoamperograms at short times after potential steps for a copper-gold alloy. Temperature, 554°C; c~u+ = 6 4 x 10 .3 M in LiCI + KCI; alloy composition c~u = 10.5 × 10 .3 tool cm-3; electrode area, 0.372 cm 2. Values of the potential step: a, 190 mV; b, 76 mV; c, 38 mV; d, 18 mV; e, - 3 8 mV; f, - 7 6 inV.
range, using Eq. (1 lb), the variation of D~°it is represented by in the alloy composition near the interface. From the thermodynamic properties of the Au-Cu system [10], the surface concentration C~u was calculated for the different overpotentials used in the experiment. Then the diffusion cott coefficients Dcu, calculated from the Cottrell law, are cott deduced those of the alloy having a concentration Cc~ from Eq. (12) (Table 1). At 402°C, in this concentration Table 1 Values of the surface concentration cbo and the equivalent surface concentration Ccu co, (Eq. (12)) during the experiments reported in Fig. 3. The interdiffusion coefficient D~°tt deduced from the Cottrell law is also given ~/mV
102 c~ u / mol c m - 3
102 Cc ucott/ mol c m - 3
1012 Dcucott/ cm 2 s - t
- 200 - 100 -50 -20 - 10 + 10 + 20 + 50 + 100 + 200
9.16 4.24 2.59 1.81 1.59 1.21 1.05 0.66 0.30 0.05
6.38 3.22 2.16 1.66 1.52 1.27 1.17 0.92 0.69 0.53
0.08 0.11 0.12 0.24 0.34 0.49 0.64 0.84 0.56 0.97
D~°t'= 0.75 × 10-12 e x p ( - a a c c u )
(16)
where the copper concentration Ccu is in moles per cubic centimetre. At short times (less than 10 - 3 s), some systematic deviations are observed: the linear dependence of 1 vs. 1 / v ~ is no longer obeyed as shown in Fig. 5. The experimental values of 1 are well above the straight line in the short time region. These perturbations cannot be attributed to the charging of the double-layer capacity. The ohmic resistance R of the electrical circuit was 0.3 ~ , and the double-layer capacity had a typical value of 50 /xF cm-2; for an electrode surface of 0.37 cm 2, the time constant of the charging process was about 6 /zs which was about 100 times shorter than the experimental observations. Moreover, Auger measurements on the Au-Cu alloy have shown that no copper segregation occurs at the alloy surface [11]. In contrast, the contribution of adsorbed species, such as the Cu ÷ ion, could lead to an additional current density during the reduction process. However, the experimental results show that the same departure occurs for the anodic current as well. Finally, we note that the departure from the Cottrell law occurred at a time scale of 1 ms, which corresponds to a mean path of the diffusing
Table 2 Analysis of chronoamperograms at short times (Fig. 4) at a temperature of 554°C. Calculation of the relevant rapid diffusion thickness 8 and the interdiffusion coefficients D~u (Eq. (13)). C~u is the bulk concentration and c~u is the surface concentration during the pulse. Calculation performed with and without taking account of the shift of the interface (Eq. (16)) r//mV
102 C~u/mol c m - 3
102 c~u/mol c m - 3
Without shift
With shift
107 8 / c m
1012 D~,,/cm 2 s - l
107 8 / e m
1012 D~u/cm 2 s-l
190 76 38 18 - 38
1.05 1.05 2.83 1.05 2.83
- 76
1.05
0.08 0.38 1.90 0.83 4.00 2.51
7.6 6.8 2.0 7.0 3.0 3.6
9.3 7. I 6.5 4.9 2.1 3.6
6.0 6.2 1.6 5.8 2.4 3.3
9.5 6.3 4.3 4.6 1.2 2.8
208
F. Lantelme and M. Chemla / Journal of Electroanalytical Chemistry 396 (1995) 203-209
atoms of about d = 2/f~D-~c,t= 10 -7 cm. During that time, the diffusion occurred only in the first external atomic layers, where diffusion may be easier than in the bulk of the alloy. The short time part of the chronoamperograms was studied by digital simulation using the RRDT concept. For given values of 6 and Dcu (see Eq. (13)), the curves were computed using the finite difference method; these calculated curves were then compared with the experimental chronoamperograms. The parameters 6 and Dcu were then changed to obtain the best fit between the experimental and calculated curves. To obtain a complete description of this system, the dependence of Dc, on the concentration and movement of the interface was also included. To detect the importance of these various contributions, we examined one experiment using, initially, the RRDT concept alone. Then the influence of the movement of the interface was taken into account. As predicted by the theory [4], the results reported in Table 2 indicate that the diffusion coefficient calculated without considering the moving boundary in addition to the distance 6 is overestimated when the concentration of copper in the surface alloy becomes large. As an example, we now give the detailed procedure used for the processing of data obtained by chronoamperometry in the Au-Cu system at 554°C. The concentration dependence of Dc. is introduced in the computations, according to our previous experimental determinations [6] Dcu = D~o exp( -53.4Cc~ )
(17)
D~, is a constant corresponding to the diffusion coefficient in the bulk of pure gold (tracer diffusion coefficient of copper) when D is assumed to be independent of the distance. The whole set of experiments reported in Fig. 5 was analysed in the same way considering the dependence of Dc. on the distance and on the concentration according to the equation Dcu = D~, exp( -53.4Ccu +
6/x)
(18)
D~, is a constant corresponding to the bulk tracer diffusion coefficient of copper in pure gold calculated by taking into account the dependence of Dc, on the distance and on the concentration. The results reported in Table 3 indicate that Table 3 Analysis of chronoamperograms at short times (Fig. 4). Results from numerical integration of Eq. (10). The interdiffusion coefficient depends on the distance x and on the concentration Ccu. Values of 8 (RRDT) and the constant D~u (Eq. (18)) are also given r//mV
102 c~u / mol c m - 3
102 c~:u / mol c m - 3
107 8 / c m
l012 D~u// cm 2 s -
190 76 38 18 -38 -76
1.05 1.05 2.83 1.05 2.83 1.05
0.08 0.38 1.90 0.83 4.00 2.51
6.6 6.0 1.5 6.1 3.2 4.1
10.7 8.9 14.7 7.2 6.8 6.9
the RRDT concept is convenient to represent the behaviour of the alloy formation or dissolution at short times; the deviation between experiment and calculation is about 7.5%. This set of experiments leads to coherent values of the thickness affected by the change in diffusion coefficient; the mean value of 6 is 4.6 X 10 - 7 cm, which corresponds to about the first ten atomic layers. Therefore all the experimental data of interdiffusion in the goldcopper system can be represented by Eq. (18) where D~u =7.5 X 10 -12 cm 2 s -j.
5. Discussion Transient electrochemical techniques are convenient and probably unique for the investigation of transport processes at very short times. Experimental results exhibit a large departure from classical laws and indicate that the rate of transport in the solid state becomes faster at short times, i.e. at very short distances from the metal surface (Fig. 5). For example, the predicted value of the concentration near the interface, as computed from diffusion equations, often leads to molar fractions X larger than unity. In fact, this erroneous conclusion can be excluded by considering the real concentration profile near the interface which is a direct consequence of the RRDT [2] (Fig. 3). Various mechanisms can be invoked to account for this observation: double-layer charging, adsorption of electroactive species, surface roughness. However, careful examination of the results leads to another interpretation, based on a net increase of the diffusion coefficient in the region close to the surface, which takes into consideration the mechanism of diffusion itself. The theories of solid state physics indicate that atomic diffusion is controlled by the concentration Nd of lattice defects. According to the Schottky theory No = NO exp( -
AGa/RT )
(19)
where NO is the frequency factor depending on the crystal structure. In the theory developed by Schottky, the energy of defect formation AGO is deduced from the summation of several terms containing mainly the coulombic interaction, the repulsion term, van der Waals' and polarization interactions and the residual energy at zero point [11]. In the classical lattice defect theory, AGO is obtained by integration of the potential terms carried out on the whole space (4rr solid angle). The important consequence of our results is that, in the initial atomic layers, the number of lattice defects should be different from that in the bulk of the solid, since the integration of potential terms is performed on a smaller solid angle. In the extreme case of the first layer, the summation is made on a 2~" solid angle and thus AG0(s) = 1/2 AGd(b). Therefore the molar fraction of defects at the surface could reach the value Nd(S) = f ~ b - ) , and could be responsible for a steep increase in the diffusion coefficient. As an example, if we assume
F. Lantelme and M. Chemla / Journal of Electroanalytical Chemistry 396 (1995) 203-209
that the defect concentration could be estimated to be 10-10 in the bulk, the concentration could reach 10 -5 near the surface. This concept is of interest in examining the electrochemical processes, as it may provide an alternative interpretation of the mechanism of the underpotential deposition of the metal. In the conventional interpretation [ 13], it was assumed that the surface activity results from a contribution of a few initial layers [14]; the thickness of this contributing region was called the relevant surface activity thickness (RSAT) by Verbrugge and Tobias [15]. The same effect can be deduced from the RRDT concept. The surface activity results from a more disordered structure of the lattice in the initial layers: it can be stated that the contribution of the inner layers arises from a dynamic effect. The enhanced formation of a surface solid solution and the metal-substrate interaction facilitate the metal deposition which occurs at a potential which is less negative than the pure metal deposition. This mechanism is illustrated in Fig. 3 where the concentration profiles calculated from the RRDT (curve a) and conventional (curve b) approaches are represented for the same surface concentration c s. The concentration CRSAT , deduced from the conventional approach, was calculated for a distance RSAT = 8; it represents the mean concentration over a distance 8, deduced from the concentration profile (curve b). In the conventional underpotential theory developed by Verbrugge and Tobias [15], this concentration is assumed to be the surface concentration; for example, for an ideal solution, it is used to calculate the electrode potential through the Nernst law. The potential change due to the shift of the concentration from CRSATto C~ expresses the apparent underpotential of the electrode
209
when its behaviour is interpreted using the conventional theory with a constant interdiffusion coefficient.
Acknowledgement The authors wish to thank Professor G. Mamantov for helpful discussions.
References [1] N.C. Cook, Sci. Am., 221 (1969) 38. [2] F. Lantelme, A. Derja and N. Kumagai, J. Electroanal. Chem., 248 (1988) 369. [3] K.B. Oldham and D.O. Raleigh, J. Electrochem. Soc., 118 (1971) 252. [4] F. Lantelme, J. Electroanal. Chem., 196 (1985) 227. [5] D.B. Butrymowicz, J.R. Manning and M.E. Read, Diffusion Data and Mass Transport, Diffusion Data Center, Metallurgy Division, Institute for Materials Research, Vol. V, INCRA Series, NBS, Washington DC, 1977. [6] F. Lantelme and M. Chemla, Z. Naturforsch., Teil A, 38 (1983) 106. [7] D. Britz, Digital Simulation in Electrochemistry, Springer-Verlag, 2nd edn., 1988. [8] M. Chemla and J. P6ri6, French Patent, N 1332 518, 1963. [9] C. Wagner, Thermodynamics of Alloys, Addison-Wesley, Reading, MA, 1952, p. 47. [10] F. Lantelme, S. Belaidouni and M. Chemla, J. Chim. Phys., 76 (1979) 423. [11] JM. McDavid and S.C. Fain, Surf. Sci., 52 (1975) 161. [12] P.G. Shewmon, Diffusion in Solids, McGraw-Hill, New York, 1963. [13] DM. Kolb, M. Prazasnyski and H. Gerischer, J. Electroanal. Chem., 54 (1974) 25. [14] Kh.Z. Brainina, N.F. Zakharchuk, D.P. Synkova and I.G. Yudelevich, J. Electroanal. Chem., 35 (1972) 165. [15] M.W. Verbrugge and C.W. Tobias, J. Electrochem. Soc., 132 (1985) 1298.
ELSEVIER
Journal of Eiectroanalytical Chemistry 396 (1995) 203-209
Chronoamperometry for the determination of metallic interdiffusion coefficients. Rapid transport processes in the first atomic layers l Fr6d6ric Lantelme, Marius Chemla Laboratoire d'Electrochimie, UniversiM Pierre et Marie Curie, 4 Place Jussieu, F 75252 Paris Cedex 05, France Received 17 January 1995; in revised form 3 February 1995
Abstract
The potential step technique was used to study the interdiffusion in the solid state at high temperature using a molten electrolyte. The method was applied to several binary metal systems and proved to be efficient for the determination of the interdiffusion coefficient D as a function of the temperature and composition of the generated alloys. The electrochemical response at short times showed a net departure from the usual diffusion laws and could be interpreted in terms of a very fast transport in the initial layers of the metal interface. Analysis of the transients was carded out by digital simulation in order to take into account the dependence of D on the concentration and distance. Keywords: Chronoamperometry; lnterdiffusion coefficient; Binary metal system
1. Introduction Electrodeposition from fused electrolytes has been used to form protective metallic coatings with good anticorrosion properties. Three electrolytic deposition processes have been developed: (a) electroplating, which produces a constant composition layer of metal coating on the substrate; (b) metalliding, which involves deposition and solid state intermetallic diffusion; (c) alloy deposition by simultaneous electroreduction of two or more cations. When the deposition is carried out at high temperature, the kinetics of formation of an intermetallic diffusion layer are fast and result in the growing of an intermediate alloy and good adhesion of the deposited coating to the substrate metal [ 1]. These technical applications of electrocoating from molten salts involve the study of electrochemical reactions which proceed under the control of the diffusion of electroactive species: diffusion of ions in the liquid electrolyte and of metallided atoms into the solid substrate. These two kinds of diffusion in the electrolyte and in the solid state are also involved when the coating is built up by the codeposition of several elements; the composition of the coating depends on the diffusion fluxes in the electrolyte and on the interdiffusion in the coating [2].
i Dedicated to Professors K. Honda, H. Matsuda and R. Tamamushi on the occasion of their 70th birthdays and in recognition of their contribution to electrochemistry. 0022-0728/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 0 2 2 - 0 7 2 8 ( 9 5 ) 0 3 9 3 3 - 3
These mechanisms can be studied quantitatively by the use of transient electrochemical techniques. The response is dependent on the time scale under consideration. Most generally, as a consequence of the simple electrochemical reactions and the high temperature, the electron transfer is fast enough to give a reversible response; then the surface activity of the deposited metal is controlled by the electrode potential. Detailed examination of the experimental results reveals that additional effects should be taken into account in order to perform a correct analysis of the transient responses. (1) There is a shift in the electrode lelectrolyte interface during the passage of the current due to metal deposition or dissolution. The influence of this effect has already been studied and mathematical analyses have been performed [3,4]. (2) The interdiffusion coefficient [5] depends on the alloy composition, which introduces a large perturbation in the analysis of transients. In a previous paper [6], we have shown that this influence does not modify the classical Cottrell relation, i.e. the transient current density j remains proportional to the reciprocal of the square root of the time t, j = A/v/-[. However, the dependence of the proportionality factor A on the diffusion coefficient and on the concentration has not yet been established. (3) At very short times, the response contains additional
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terms, such as double-layer capacity charging or electroreduction of adsorbed species. From our present study, we observed that in most experiments the current density is enhanced with respect to the predicted values at very short times. This result could be assigned to a rapid motion of the deposited atoms in the initial layers of the interface. This article is devoted to the analysis of the transient response, which includes the influence of these various phenomena that are not included in the conventional treatment. To obtain a better understanding of this behaviour, we present first a description of the experimental procedure. These transport problems cannot be solved algebraically, and therefore in order to obtain fundamental information, the electrochemical results are analysed using the digital simulation method [7].
2. Experimental The method involves the use of polarization cells which have the general configuration Alloy electrode Liquid electrolyte I Pure M M-M' containing Mn + ,
or metal atoms M) in the bulk of the two phases (electrolyte and metal) obey the equation
OCj( x, t) 0 I Cj( x, t) l at = 7x Dj o-----T--
(2)
where Dj and cj are the diffusion coefficient and concentration of the electroactive species respectively, and x is the distance normal to the electrode. The initial boundary condition is
q ( x, 0) = c 3"
(3)
c[ is the initial concentration of the electroactive species. The thickness of the diffusing medium is considered to be large compared with the diffusion layer; the condition of semi-infinite diffusion is then fulfilled cj(o¢, t) = cj"
(4)
The flow of metal from the surface into the bulk of this electrode by diffusion is limited by the flux of metal J formed by electrodeposition and equal to it. Therefore the current density j associated with the perturbation is (j being positive for an oxidation process) ~x
j = - nFJ = - nFD i
x= 0
(5)
(1)
M is the electroactive metal and M' is the metal substrate. In most of our experiments, the electrolyte was a eutectic LiC1 + KCI mixture whose component salts were obtained from Merck (analytical reagen0. Before each experiment, the mixture was heated in the cell at 130°C under vacuum for 4 h, and then heated at the experiment temperature (400-540°C) while bubbling HCI for 6 b. Finally, argon gas was circulated for a few hours to eliminate HCI. The metal ions M were generated by the anodic dissolution of a rod of pure M; the counterelectrode was a chlorine electrode [8] placed in a separate compartment connected with a porous tube. The chlorine electrode was also used as a reference electrode. For the measurements using cell (1), the secondary reference electrode and the counterelectrode were made of a rod of pure M. The working electrode was a wire, I mm in diameter, of Au-Cu alloy. An electronic device built in our laboratory was used to deliver potentiostatic pulses. The potential pulses and current responses were recorded on a double-trace digital oscilloscope Nicolet 310. The purity of the molten electrolyte was examined by cyclic voltammetry. The residual current density measured with a platinum electrode over the potential range -0.5, - 1 . 8 V vs. CI 2 IC1- was less than 0.1 m A c m -2.
which is available for both diffusion of M n+ in the liquid phase and of M atoms in the solid metal substrate (Fig. 1). The concentration profile c j ( x , t) is calculated by integrating the diffusion equation (Eq. (2)) with boundary conditions expressing the electrochemical constraints arising from the experimental conditions. For example, using the potentiostatic method, the variation of the current density vs. time should obey the Cottrell law, provided that the diffusion constant D is a constant, following the expression j = n F ( cj - cj" ) ~ j /
(6)
Trt
This expression is valid for the determination of Dj in the liquid phase and in the solid metal [6]. In fact, the
etal• c
CsM
Solid m
C"M
Liquid
phase
C'Mn+
CSMn+
Distance
3. Basic principle of the method When planar diffusion is the rate-determining step, the concentration changes of electroactive species j (ions, M "+
x
Fig. 1. Profile of the concentrations of electroactive species during electrochemical reduction in both the electrolyte and the solid substrate phase.
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F. Lantelme and M. Chemla/ Journal of Electroanalytical Chemistry 396 (1995) 203-209
concentration changes in the metal and the electrolyte are interrelated at any time by CM -- CM / DM"+ "+ -- CM,+ * = V OM cM
(7)
The diffusion coefficient in the solid metal is generally smaller by a factor of 10 6 compared with the melt, and the potential step is easily adjusted to appropriate values to make the current only dependent on the solid state interdiffusion; the difference CM,+-- Cry,+ is very small, and thus the values of c M and c M are determined from the Nernst law by the values of the imposed potentials provided that the activity coefficients of the components are known. In most cases, thermodynamic data are available in the literature and may be expressed by a Margules [9] expansion In T - a 2 / 2 ( 1 - X ) z + a 3 / 3 ( 1 - - X ) 3 + a4/4(1
--X)4 -[- ...
(8)
where X is the molar fraction of species j. The potential step can be chosen to be either positive or negative with respect to the equilibrium value in order to measure Dj when diffusion takes place in or out of the sample material. Most generally, for a given composition, we observed that the experimental Dj values do not depend on whether the diffusion takes place in or out of the substrate. Moreover, the variation of the amplitude of the potential perturbation permits surface alloys to be generated with different compositions. The method therefore provides an efficient tool for the determination of D M values as a function of concentration. Three mathematical difficulties appear in the solution of the diffusion equation (Eq. (2)) corresponding to a given technique. (1) The diffusion coefficient of the deposited metal may be quite different from that of the substrate. This property induces a Kirkendall effect and leads to a movement of the internal reference frame. Only an interdiffusion coefficient Dj is then observed. (2) The interdiffusion coefficient of the metal is strongly dependent on the composition of the alloy; thus a variable Dj should be introduced along the concentration gradient. (3) The deposition of the metal results in a shift of the interface which must be accounted for as a moving origin of the abscissae. For these conditions, the conventional equations of chronoamperometry and chronopotentiometry are not applicable. New mathematical expressions have been developed to account for the moving boundary [4]. However, when perturbing phenomena, such as the concentration and distance dependence of Dj, are superimposed on the condition of the moving boundary, the problem is not solvable, Digital simulation is then the only way to treat the phenomena comprehensively. The outline of the implementation of this technique is described below.
It is assumed that the diffusion coefficient Dj depends on both the concentration cj of the active species and the distance x to the interface. The rate of movement of the interface (9)
v =j/nFVj
is also taken into account [4]; Vj is the partial molar volume of the electroactive species. For a metallic solid solution, the sign of the current density is positive when the electroactive metal is dissolved (oxidation), i.e. when the interface moves towards the bulk of the sample. Then the diffusion equation (Eq. (2)) is written as 0cj( x, t) ~t
=
aDj 0cj ~x ~x
- - - -
+
02cJ OJ - ~ x2
+
v ~cj -0x
(lO)
Dj depends on x and cj which is also dependent on x; thus d Dj
ODj --
dx
_
_
~x
aDj Ocj -[-
-
-
0cj ax
Eq. (10) is solved by the technique of numerical simulation using the point method [7]. The time step At is a fraction of the total duration t T of the experiment (At ~/T/10000). According to the usual convergence requirements [7], the space interval was chosen to be equal to: /ix = 2.1~LZ1"~; D L is the largest diffusion coefficient involved in the experiment. Naturally, the convergence of the technique was carefully checked using mesh refinement methods. We now examine separately the two cases where Dj depends either on the concentration or on the distance to the interface, the shift of the interface being neglected. The interpretation of experiments where Dj depends on both the concentration and distance will be examined later. 3.1. Dependence o f D on concentration
Two cases are considered: a linear dependence Dj = Dj°(1 + hcj)
(lla)
and an exponential dependence Oj = Oi ° exp(hcj)
(1 lb)
where Dj° is the limiting value of the diffusion coefficient at zero c and h is a constant parameter. The numerical treatment of Eq. (10) indicates that the current always remains proportional to the reciprocal of the square root of time as in the Cottrell law. This property of the potentiostatic method has been previously demonstrated using the Boltzmann transformation [6], since the limiting conditions of the method can be expressed with a non-dimensional variable x/D~'fl. The linear relationship of j vs. I / ~ is valid even if Dj is dependent on c and regardless of the variation of Dj as a function of cj [6]. The slope of the linear variation of j vs. the reciprocal of the square root of t was used to determine a particular value of the diffusion
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F. Lantelme and M. Chemla/ Journal of Electroanalytical Chemistry396 (1995) 203-209
coefficient, D ; °tt, simply by applying the Cottrell equation (Eq. (6)). The numerical treatment indicated that D; °tt is a value of Dj deduced from Eq. (lla) or (lib) for a concentration, C; °tt, n e a r the interface. Assuming that the variations of the diffusion coefficient are not too large, this concentration was computed to be approximately equal to C; °tt~" 0 . 6 4
c~ + 0.36cj*
:C......Z' C'- C' 0,8
0.6
(12) 0.4
where c~ is the surface concentration and cj* is the bulk concentration.
.C1~I
_ - -
0.2
3.2. Dependence of D on distance 2
This dependence has been studied recently to take into account the results of transient techniques when the observation time is short enough to detect the transport phenomena in the initial atomic layers near the interface [2]. It has been proposed that the increase in transport kinetics in this region can be represented by introducing a variable diffusion coefficient which increases as the distance to the interface decreases. Here, as in our previous article [2], it was assumed that Dj obeys the equation Dj
=
Dj* e x p ( 8 / x )
(13)
where D[ is the bulk interdiffusion coefficient and 8 is a characteristic distance called the relevant rapid diffusion thickness (RRDT) [2]. By introducing Eq. (13) into Eq. (2), the diffusion equation to be solved is 0cj(x, t) 8 0Cj 02Cj at = Dj'~-~""~" X +Dj ax 2
(14)
At short distances, Dj becomes very large; thus, in this region, the concentration profile is nearly flat. For the numerical calculation, it was considered that, at a distance shorter than 8' = p8 (with p = 0.25), the concentration c remains constant. The numerical computations show that a Cottrell-type relationship is obeyed only at long times. Departures appear at short times: the ratio between the calculated current density j and the current density Jc deduced from the Cottrell law increases as the time t decreases. The time
I/J¢ 1.5
1.4
1.3
,
,
,
,
0.5 •
.
.
.
.
.
1;o
.
.
1.,~/~ .
Fig. 2. Departure from the Cottrell law, J/Jc vs. l / ~ / ~ ' s ) , j is the calculated current density taking account of the rapid diffusion in a layer of thickness d (RRDT). j~ is the current density deduced from the Cottrell law. Characteristic time of the system, ~'c = ~$2 / 2 Dj °.
,
x/6
Fig. 3. Concentration profile near the alloy surface: (a) calculated from the RRDT concept introducing a variation of the diffusion coefficient near the interface (Eq. (13)); (b) deduced from the classical treatment for
a constantdiffusioncoefficient Dj°. The mean concentration CRSATis calculated for a thickness(RSAT)equal to 8 (Eq. (1 la)). scale at which this departure becomes significant is characterized by the time ~s
rs = 82/2Dj
(15)
The variations in the ratio J/Jc vs. 1 / t ~ ) obtained by numerical computation are reported in Fig. 2. The Cottrell law is only obeyed at large t; for t = 100r~, the departure is about 8%. The RRDT concept has an important consequence for the distribution of electroactive species within the initial surface layers. Indeed, when x becomes small, we observe from the numerical treatment that the slope of the concentration profile tends towards zero, even for small t, in contrast with the value (cj* - c~/~V/-~--Djt, predicted by the classical theory. As indicated in Fig. 3 (curve a), the profile remains very flat near the interface, whereas the conventional calculation indicates a steep concentration change in this region (curve b).
4. Interdiffusion in the copper-gold system Using the basic principles developed above, we examine the results of experiments carried out by the potential step method to study the interdiffusion of copper and gold using cell (1) where M = Cu and M' = Au [6]. The electrolyte was a fused eutectic LiC1 + KC1. The copper ions Cu ÷ were introduced into the bath by the anodic dissolution of a pure copper rod (Johnson-Matthey). The transient currents l(t) observed at a scale of 10 -2 s, after a steep change of the potential, are shown in Fig. 4. The plots in Fig. 4 are in agreement with the theoretical result of a linear dependence of 1 vs. 1/ v~" even when Dj is dependent on c. However, as computed by our numerical treatment, the diffusion coefficients deduced from the Cottrell equation (Eq. (6)) showed a strong dependence on the overpotential r/. This effect corresponds to the change
F. Lantelme and M. Chemla / Journal of Electroanalytical Chemistry. 396 (1995) 203-209 limA
•"
I/~A
207
J ,40 .20
5
• 20 -40 -60
20
Fig. 4. Analysis of chronoamperograms after potential steps for a copper-gold alloy (Eq. (4)). Temperature, 402°C; c~u+ =91 × 10 -3 M in LiC1 + KCI; alloy composition c~u = 13.7 X i 0 - 3 tool c m - 3; electrode area, 1.I 1 cm 2. Values of the potential step: a, a', 200 mV; b, b', 100 mV; c, c', 50 mV; d, d', 20 mV; e, e', 10 mV.
4
0
,t"It21 S.
Fig. 5. Analysis of c~onoamperograms at short times after potential steps for a copper-gold alloy. Temperature, 554°C; c~u+ = 6 4 x 10 .3 M in LiCI + KCI; alloy composition c~u = 10.5 × 10 .3 tool cm-3; electrode area, 0.372 cm 2. Values of the potential step: a, 190 mV; b, 76 mV; c, 38 mV; d, 18 mV; e, - 3 8 mV; f, - 7 6 inV.
range, using Eq. (1 lb), the variation of D~°it is represented by in the alloy composition near the interface. From the thermodynamic properties of the Au-Cu system [10], the surface concentration C~u was calculated for the different overpotentials used in the experiment. Then the diffusion cott coefficients Dcu, calculated from the Cottrell law, are cott deduced those of the alloy having a concentration Cc~ from Eq. (12) (Table 1). At 402°C, in this concentration Table 1 Values of the surface concentration cbo and the equivalent surface concentration Ccu co, (Eq. (12)) during the experiments reported in Fig. 3. The interdiffusion coefficient D~°tt deduced from the Cottrell law is also given ~/mV
102 c~ u / mol c m - 3
102 Cc ucott/ mol c m - 3
1012 Dcucott/ cm 2 s - t
- 200 - 100 -50 -20 - 10 + 10 + 20 + 50 + 100 + 200
9.16 4.24 2.59 1.81 1.59 1.21 1.05 0.66 0.30 0.05
6.38 3.22 2.16 1.66 1.52 1.27 1.17 0.92 0.69 0.53
0.08 0.11 0.12 0.24 0.34 0.49 0.64 0.84 0.56 0.97
D~°t'= 0.75 × 10-12 e x p ( - a a c c u )
(16)
where the copper concentration Ccu is in moles per cubic centimetre. At short times (less than 10 - 3 s), some systematic deviations are observed: the linear dependence of 1 vs. 1 / v ~ is no longer obeyed as shown in Fig. 5. The experimental values of 1 are well above the straight line in the short time region. These perturbations cannot be attributed to the charging of the double-layer capacity. The ohmic resistance R of the electrical circuit was 0.3 ~ , and the double-layer capacity had a typical value of 50 /xF cm-2; for an electrode surface of 0.37 cm 2, the time constant of the charging process was about 6 /zs which was about 100 times shorter than the experimental observations. Moreover, Auger measurements on the Au-Cu alloy have shown that no copper segregation occurs at the alloy surface [11]. In contrast, the contribution of adsorbed species, such as the Cu ÷ ion, could lead to an additional current density during the reduction process. However, the experimental results show that the same departure occurs for the anodic current as well. Finally, we note that the departure from the Cottrell law occurred at a time scale of 1 ms, which corresponds to a mean path of the diffusing
Table 2 Analysis of chronoamperograms at short times (Fig. 4) at a temperature of 554°C. Calculation of the relevant rapid diffusion thickness 8 and the interdiffusion coefficients D~u (Eq. (13)). C~u is the bulk concentration and c~u is the surface concentration during the pulse. Calculation performed with and without taking account of the shift of the interface (Eq. (16)) r//mV
102 C~u/mol c m - 3
102 c~u/mol c m - 3
Without shift
With shift
107 8 / c m
1012 D~,,/cm 2 s - l
107 8 / e m
1012 D~u/cm 2 s-l
190 76 38 18 - 38
1.05 1.05 2.83 1.05 2.83
- 76
1.05
0.08 0.38 1.90 0.83 4.00 2.51
7.6 6.8 2.0 7.0 3.0 3.6
9.3 7. I 6.5 4.9 2.1 3.6
6.0 6.2 1.6 5.8 2.4 3.3
9.5 6.3 4.3 4.6 1.2 2.8
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F. Lantelme and M. Chemla / Journal of Electroanalytical Chemistry 396 (1995) 203-209
atoms of about d = 2/f~D-~c,t= 10 -7 cm. During that time, the diffusion occurred only in the first external atomic layers, where diffusion may be easier than in the bulk of the alloy. The short time part of the chronoamperograms was studied by digital simulation using the RRDT concept. For given values of 6 and Dcu (see Eq. (13)), the curves were computed using the finite difference method; these calculated curves were then compared with the experimental chronoamperograms. The parameters 6 and Dcu were then changed to obtain the best fit between the experimental and calculated curves. To obtain a complete description of this system, the dependence of Dc, on the concentration and movement of the interface was also included. To detect the importance of these various contributions, we examined one experiment using, initially, the RRDT concept alone. Then the influence of the movement of the interface was taken into account. As predicted by the theory [4], the results reported in Table 2 indicate that the diffusion coefficient calculated without considering the moving boundary in addition to the distance 6 is overestimated when the concentration of copper in the surface alloy becomes large. As an example, we now give the detailed procedure used for the processing of data obtained by chronoamperometry in the Au-Cu system at 554°C. The concentration dependence of Dc. is introduced in the computations, according to our previous experimental determinations [6] Dcu = D~o exp( -53.4Cc~ )
(17)
D~, is a constant corresponding to the diffusion coefficient in the bulk of pure gold (tracer diffusion coefficient of copper) when D is assumed to be independent of the distance. The whole set of experiments reported in Fig. 5 was analysed in the same way considering the dependence of Dc. on the distance and on the concentration according to the equation Dcu = D~, exp( -53.4Ccu +
6/x)
(18)
D~, is a constant corresponding to the bulk tracer diffusion coefficient of copper in pure gold calculated by taking into account the dependence of Dc, on the distance and on the concentration. The results reported in Table 3 indicate that Table 3 Analysis of chronoamperograms at short times (Fig. 4). Results from numerical integration of Eq. (10). The interdiffusion coefficient depends on the distance x and on the concentration Ccu. Values of 8 (RRDT) and the constant D~u (Eq. (18)) are also given r//mV
102 c~u / mol c m - 3
102 c~:u / mol c m - 3
107 8 / c m
l012 D~u// cm 2 s -
190 76 38 18 -38 -76
1.05 1.05 2.83 1.05 2.83 1.05
0.08 0.38 1.90 0.83 4.00 2.51
6.6 6.0 1.5 6.1 3.2 4.1
10.7 8.9 14.7 7.2 6.8 6.9
the RRDT concept is convenient to represent the behaviour of the alloy formation or dissolution at short times; the deviation between experiment and calculation is about 7.5%. This set of experiments leads to coherent values of the thickness affected by the change in diffusion coefficient; the mean value of 6 is 4.6 X 10 - 7 cm, which corresponds to about the first ten atomic layers. Therefore all the experimental data of interdiffusion in the goldcopper system can be represented by Eq. (18) where D~u =7.5 X 10 -12 cm 2 s -j.
5. Discussion Transient electrochemical techniques are convenient and probably unique for the investigation of transport processes at very short times. Experimental results exhibit a large departure from classical laws and indicate that the rate of transport in the solid state becomes faster at short times, i.e. at very short distances from the metal surface (Fig. 5). For example, the predicted value of the concentration near the interface, as computed from diffusion equations, often leads to molar fractions X larger than unity. In fact, this erroneous conclusion can be excluded by considering the real concentration profile near the interface which is a direct consequence of the RRDT [2] (Fig. 3). Various mechanisms can be invoked to account for this observation: double-layer charging, adsorption of electroactive species, surface roughness. However, careful examination of the results leads to another interpretation, based on a net increase of the diffusion coefficient in the region close to the surface, which takes into consideration the mechanism of diffusion itself. The theories of solid state physics indicate that atomic diffusion is controlled by the concentration Nd of lattice defects. According to the Schottky theory No = NO exp( -
AGa/RT )
(19)
where NO is the frequency factor depending on the crystal structure. In the theory developed by Schottky, the energy of defect formation AGO is deduced from the summation of several terms containing mainly the coulombic interaction, the repulsion term, van der Waals' and polarization interactions and the residual energy at zero point [11]. In the classical lattice defect theory, AGO is obtained by integration of the potential terms carried out on the whole space (4rr solid angle). The important consequence of our results is that, in the initial atomic layers, the number of lattice defects should be different from that in the bulk of the solid, since the integration of potential terms is performed on a smaller solid angle. In the extreme case of the first layer, the summation is made on a 2~" solid angle and thus AG0(s) = 1/2 AGd(b). Therefore the molar fraction of defects at the surface could reach the value Nd(S) = f ~ b - ) , and could be responsible for a steep increase in the diffusion coefficient. As an example, if we assume
F. Lantelme and M. Chemla / Journal of Electroanalytical Chemistry 396 (1995) 203-209
that the defect concentration could be estimated to be 10-10 in the bulk, the concentration could reach 10 -5 near the surface. This concept is of interest in examining the electrochemical processes, as it may provide an alternative interpretation of the mechanism of the underpotential deposition of the metal. In the conventional interpretation [ 13], it was assumed that the surface activity results from a contribution of a few initial layers [14]; the thickness of this contributing region was called the relevant surface activity thickness (RSAT) by Verbrugge and Tobias [15]. The same effect can be deduced from the RRDT concept. The surface activity results from a more disordered structure of the lattice in the initial layers: it can be stated that the contribution of the inner layers arises from a dynamic effect. The enhanced formation of a surface solid solution and the metal-substrate interaction facilitate the metal deposition which occurs at a potential which is less negative than the pure metal deposition. This mechanism is illustrated in Fig. 3 where the concentration profiles calculated from the RRDT (curve a) and conventional (curve b) approaches are represented for the same surface concentration c s. The concentration CRSAT , deduced from the conventional approach, was calculated for a distance RSAT = 8; it represents the mean concentration over a distance 8, deduced from the concentration profile (curve b). In the conventional underpotential theory developed by Verbrugge and Tobias [15], this concentration is assumed to be the surface concentration; for example, for an ideal solution, it is used to calculate the electrode potential through the Nernst law. The potential change due to the shift of the concentration from CRSATto C~ expresses the apparent underpotential of the electrode
209
when its behaviour is interpreted using the conventional theory with a constant interdiffusion coefficient.
Acknowledgement The authors wish to thank Professor G. Mamantov for helpful discussions.
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