Electrochemical Deposition of Metals Inside High Aspect Ratio

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J. Phys. Chem. C 2007, 111, 5229-5235

5229

Electrochemical Deposition of Metals Inside High Aspect Ratio Nanoelectrode Array: Analytical Current Expression and Multidimensional Kinetic Model for Cobalt Nanostructure Synthesis L. Philippe,* N. Kacem, and J. Michler EMPA (Swiss Federal Laboratories for Materials Testing and Research), Feuerwerkerstrasse 39, 3602 Thun, Switzerland ReceiVed: NoVember 23, 2006; In Final Form: February 6, 2007

We report for the first time a comprehensive kinetic model based on the coupling of a cylindrical ultrarecessed electrode (Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M. J. Electroanal. Chem. 1987, 217, 417.) and the fractional area: this takes into account the derivation of a global diffusion coefficient in the steady-state current regime and considers the geometry of the nanoelectrode influence on the process. An experimental validation is done with the electrochemical deposition of cobalt inside a tracketched polymer membrane. This shows the kinetic dependence over the concentration, overvoltage, and geometrical parameters of the system. A decoupling of those parameters on the diffusion term is done, and a semianalytical model was derived for the case of cobalt. An estimation of the transference number and heterogeneous rate constant of the process is experimentally derived. This model provides the possibility of the prediction of electrosynthesis conditions with a quantitative evaluation of determining factors for the resulting nanostructure.

Introduction Nanostructured materials are of considerable interest for potential applications in electronic, optical, and magnetic devices because of their peculiar structural characteristics, size/shape effects, and physical properties that are different from the bulk materials. The electrochemical process for nanostructure synthesis has the advantage over the other techniques to create a composite structure within the pores and to tune nanostructure properties by changing the synthesis parameters. Compared with chemical vapor deposition (CVD), molecular-beam epitaxy (MBE), and sputtering techniques, the advantages of electrodeposition over the mentioned vacuum-based techniques include high deposition rates, a simple experimental setup, and low cost. Additionally, the electrodeposition techniques remove the restriction of high heat resistance of the substrate and the deposition can be carried out at room temperature. These advantages explain easily the success of the method where reliable size-controlled nanorods and nanowires can be produced.2 Chemically synthesized templates such as liquid crystal template mesoporous aluminosilicates,3 electrochemically synthesized templates such as anodically etched aluminum4 or silicon,5 or track-etched polymer membranes6 can all be used to produce electrodeposited nanostructures. A great deal of reports on the synthesis of semiconductors,7 metals,8 and heteronanostructures9 are available in the literature. There is growing interest in metallic nanowires, especially magnetic because of their wide range of promising applications in various fields, which include high-density data storage, ferrofluidic, and magnetic resonance imaging (MRI). On that specific domain of research, the influence of the synthesis parameters (metallic ion concentration, bath composition, pH solution, or polarization mode) on the nanowire’s structure and magnetic properties have * Corresponding author. Phone +41(0)332285249. Fax +41(0)332284490. E-mail: [email protected].

been reported experimentally.10,11 More generally, because it is clearly suspected that the electrochemical deposition mechanism inside the templated substrate is a determining factor for the control and tuning of the properties of electrodeposited nanostructures, it is of great need to develop an understanding of such mechanisms in depth. As an electrode is miniaturized, the following phenomena are observed: the mass transport of the electroactive species is varied from the linear diffusion normal to the electrode surface to the two- or three-dimensional diffusion, the current becomes smaller but is proportional to the electrode area, and the current density decreases.12,13 The advantages are that a steady-state current is obtained and allows making chemical and electrochemical kinetic measurements and arrangement of ultramicroelectrodes (UME) provides new functionality owing to reactions of electroactive species. Historically, the method of electrochemical synthesis of nanostructures using templates was introduced by Possin.14 At present, although this process is used widely, little is known about the mechanism itself for which some similarities to the behavior of UME ensembles can be expected. Scho¨nenberger et al.15 reported the diffusion-limited process influence by the geometry of the structures, and Schuchert et al.16 propose a qualitative model of the process based on experimental evidence. A kinetic model is derived only for the short time period of the charge-transfer controlled regime. Recently, Valizadeh et al.17 proposed a quantitative consideration of concentration profiles during electrodeposition of a cobalt nanowire array. A diffusion coefficient for cobalt deposition is derived by using the Cottrell equation at short times where the diffusion is linear and time-dependent. However, the growth process is characterized by the spherical diffusion regime when the steady-state current density is achieved. The aim of the work is to develop an analytical expression for the diffusion-controlled limiting current for an array of nanoelectrodes as well as to

10.1021/jp0677997 CCC: $37.00 © 2007 American Chemical Society Published on Web 03/10/2007

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Philippe et al. • At short time (region 1), the concentration gradient in front of the cathode (Nernst layer) is growing into the electrolyte but remains smaller than the remaining pore length and the diffusion is linear inside the pores. The current decreases because of the mass transport process initiation. • Then, a spherical diffusion for the longer time when the steady-state current density is achieved appears (region 2). This is because the depletion zone exceeds the pore length and a radial diffusion field establishes around each pore; the ions diffuse radially toward the pore openings. An SEM picture in Figure 1b shows a template cross section during the deposition process. • Finally, the increase of the current in region 3 indicates that caps start to overgrow the membrane surface (Figure 1c). The initiation of the current rise defines the process duration. Figure 1d shows an array of cobalt nanowires at the end of the process after the membrane used as the template was chemically dissolved. In the case of a cylindrical UME, which involves a single dimension of diffusion, a practical approximation reported by Szabo et al.1 to be valid within 1.3% is

I(t) )

nFπ r 2D(Cb - Cm) × r0

[

2 exp(-0.05π1/2τ1/2)

Figure 1. (a) Current-time curve for deposition at η ) -1 V, 1 M cobalt solution, and an average pore radius, r0, of 10 nm. Regions 1-3 define three main events on the plot: Region 1 represents the beginning of a linear diffusion inside the pores as soon as charge-transfer reaction has taken place. Region 2 shows the period when a radial diffusion is the dominating process and that a current limiting plateau is reached. Region 3 represents the current rise as caps start to overgrow the membrane surface. Here the linear diffusion is again the dominating process as in region 1. Figures 1b-d display, respectively, SEM pictures of the membrane cross section with growing nanowire inside (region 2), top view of the membrane surface with caps growing (region 3), and nanowire arrays after membrane dissolution.

derive a kinetic model of the mass transport process, which is changing as time elapses. A systematic study of the multidimensional diffusion mechanism for modeling the synthesis process of nanowires is reported and discussed. The case of cobalt deposition inside a nanoelectrode template illustrates the influence of geometry, concentration of metallic species, and current density on the process kinetics. We show the interest of the model to predict pH and concentration gradient in solution as time elapses and derive a heterogeneous rate constant for cobalt deposition depending on the electrode geometry. Theoretical Considerations Let us consider an array of nanoelectrodes on which a conductive back-side layer was sputtered. The geometry of a single nanoelectrode is shown in Figure 1 along with a current versus time plot recorded during a deposition process. This typical plot displays three main regions describing qualitatively the series of events during the deposition. At very short time, the concentration gradient is nearly nil so that the charge transfer determines the process. With increasing time, the depletion zone in front of the cathode (Nernst layer) is growing into the electrolyte. The concentration gradient is increasing, and the overall reaction starts to be diffusion-controlled:

π1/2τ1/2

+

]

1 (1) ln(5.2945 + 0.7493τ1/2)

where τ ) (4Dt/r20). For a long time, when τ becomes very large, the current becomes

I(t) )

2nFAD(Cb - Cm) r0 ln τ

(2)

The interface short-long time is characterized by the transition between a linear and spherical diffusion regime (interface between regions 1 and 2 in Figure 1). Using the fractional area (Avrami theorem)17,18 and by taking into account the concentration gradient, we obtain two forms of the fractional area for short and long time: for 0 e t e ts

{ [ [

A ) Seq 1 - exp -

NπCbxπD r0 r (4L + π r0)(Cb - Cm)

for ts e t e ∞

A ) Seq

{ [ [

r0 NπCbxπD 1 - exp r (4L + π r0)(Cb - Cm)

] ]} 1/4

t1/8

] ]}

(3)

1/2

t1/4

(4)

where ts is short time and the fractional area is defined as A ) Seq[1 - exp(-ktn)] with Seq ) π r 2. Kinetic Coupling. A form of the current where it is governed by both mass transfer and charge transfer kinetics for a simple electrode reaction is19

I(t) ) I(0) exp(λ2) erfc(λ)

(5)

where I(0) is the true exchange current. For short time (0 e t e ts)

[

I(λ) ) I(0) 1 -

2λ π1/2

]

(6)

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For long time (ts e t e ∞)

I(λ) )

I(0)

(7)

λπ1/2

For short time, using eqs 2, 3, and 6, we can deduce λ:

λ)

[

]

2nFAD(Cb - Cm) π1/2 12 r0 ln τ

(8)

The current form, at short time, holds the final expression

I(t) ) I(0) exp

{[

]}

2nFAD(Cb - Cm) 2 π 14 r0 ln τ 1/2 2nFAD(Cb - Cm) π erfc 12 r0 ln τ

{ [

]}

(9)

For long time, using eqs 2, 4, and 7, we deduce λ:

λ)

r0 I(0) ln τ

(10)

2nFADπ1/2(Cb - Cm)

The current form, at long time, holds the final expression

I(t) ) I(0) exp

{[

r0 I(0) ln τ 1 π 2nFAD(Cb - Cm) erfc

{

]}

in solution. The radial diffusion depends on the pore size, and the electrostatic diffusion is a term describing the influence of the overvoltage over the total diffusion coefficient measured. Concentration Gradient. The concentration of cobalt at the pore mouth, Cm, is calculated as follows:

2

r0 I(0) ln τ

2nFADπ1/2(Cb - Cm)

}

Cm ) (11)

Determination of the Diffusion Coefficient, D. The diffusion coefficient is derived from the current plot at long time (i.e., mass transfer controlled region):

r0

Figure 2. (ln[D]/Db) as a function of the bulk concentration Cb for different pore sizes. The inset graph shows the plot of the inverse current (eq 12) vs ln[4t/r20] in order to deduce b (intersection between the line and vertical axis ∆: x ) 0).

{

[ ]}

4t 1 ln D + ln 2 ) I(t) 2nFAD(Cb - Cm) r0

(12)

By plotting the reciprocal current versus ln[4t/r20], a linear regression is performed to extract the value of b ) (1/I(t ) r20/4)) as can be shown in Figure 2b.

ln D 2nbFA(Cb - Cm) ) D r0

(13)

The diffusion coefficient is then deduced graphically (Figure 2a) or numerically by solving the mathematical equation E(x): (ln x/x) ) Cste. The global diffusion, D, is deconvoluted into three components that represent the multidimensional diffusion growth regime. Linear and radial diffusion terms are describing the diffusion inside, above (DL), and around the pores (Dr0). Both terms being convoluted with the electrostatic diffusion term Dη, the analytical expression is as follows: 2 1/4 2 D ) D1/2 η (DL + Dr0)

(14)

DL ) Linear - Diffusion Dr0 ) Radial - Diffusion Dη ) Electrostatic - Diffusion Experimentally, the deconvolution supposes that the linear diffusion is only dependent on the metallic species concentration

R ln[τ + 1] 1 - exp[-β t1/2]

(15)

where R and β are

R)

β)r

I(0)

(16)

Nr0(π r)2

[ ] Nπ(πD)1/2 4Lt + π r0 TP

1/2

(17)

To calculate β in eq 17, the process duration, TP , representing the time needed until at least one nanoelectrode is filled up to the top surface of the membrane is needed. Experimentally, TP represents on the current-time plot the beginnning of region 3 where the current rise indicates a sudden change in the active surface area. Equation 18 proposes an analytical form TP , which takes into account the three main components influencing the growth kinetics. There are, namely, the concentration of metallic species in the bulk solution, the potential applied to the system, and the cylindrical nanoelectrode’s effective volume Veff: 2 1/2 2 (T P,C + T P,V )1/4 TP ) T P,η eff

(18)

TP,η ) Timekinetic1 TP,C ) Timekinetic2 TP,Veff ) TimeGeometry Because we assume that the process is completely diffusional in region 2, we approximate a linear dependence between the electrodeposition duration and the multidiffusional expression (eq 14). pH Gradient. During the growth process, we consider the transfer of metallic ions (Mn+) from the bulk solution into the ionic metal lattice M (deposition) and the cathodic evolution of

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Philippe et al.

TABLE 1: Average Parameter Values Used for the Experiments and Comparison of Diffusion Coefficient Values Obtained by Two Different Methods for the Case of Deposition at an Overvoltage of -1 V and a Bulk Concentration of 1 Ma pore radius r0 (nm) membrane thickness L (µm) pore density N (pores‚cm-2) diffusion coefficient using eq 12 (cm2s-1) diffusion coefficient using eq 23 (cm2s-1)

10 5 1 × 109 2 × 10-7

15 5 8 × 108 5 × 10-7

50 6 2 × 108 2 × 10-6

8 × 10-6

2 × 10-6

2 × 10-5

a Average pore sizes and density were determined for each experiment.

hydrogen in the medium as the main reactions. Equation 19 is the Nernst equation for the potential variation (En+ M /M) with concentration. The concentration gradient of the metallic ions in solution can be written as the difference between the concentration of the metallic species in the bulk solution [Mn+ b ] and the one at the pore mouth [Mn+ m ]

E(Mn+/M) ) E0M +

RT ln{[Mn+]b - [Mn+]m} nF

(19)

where E0M is the standard electrode potential of the RedOx couple and [Mn+ m ] is obtained using eq 15. By considering the bulk pH (pHb) as the pH of the bulk solution, we can deduce the pH difference existing between the pore mouth and the bulk solution:

E(H+/H2) )

0 RT ln{10-2[pHb+pH ]} 2F

(20)

By using eqs 19 and 20, we can deduce the overall reaction potential, E:

{

}

[Mn+]b - [Mn+]m 2 RT E ) E(Mn+/M) - E(H+/H2) ) E0M + ln 0 n nF 10-2[pHb+pH ] (21) On the basis of the above assumptions, an expression of the pH difference existing between the pore mouth and the bulk solution is derived:

{

}

[Mn+]b - [Mn+]m ln nF exp (E - E0M) RT 0 + pHb| |pH | ) | 2 ln[10]

[

]

Figure 3. Experimental and theoretical current-time plots for different pore sizes.

convection. A saturated-calomel electrode (SCE) was used as the reference electrode. Using a conventional potentiostat, the current is measured during electroplating at determined overpotentials η, defined as η ) Eapplied - EOCP . The following electrolyte was used for the cobalt solution: CoSO4 (1 M, 0.5 M, and 0.1 M), H3BO3 0.7 M, NaCl 0.11 M. The solution pH is adjusted to pH 3 using solution of H2SO4 or KOH. Results Current Model Validation. Figure 3 displays the experimental current-time plots measured during the potentiostatic polarization at the overpotential of -1 V, with a cobalt concentration of 1 M and for three nanoelectrode pore sizes. The dashed curves represent the current plot obtained by using eq 12. A good agreement is found with the experimental current for the three main regions of the current plot describing the changes of the regimes as time elapses. This therefore confirms the validity of the model for predicting quantitatively the current variation during a deposition process as a function of the nanoelectrode geometry. Diffusion Coefficient Derivations. Table 1 displays diffusion coefficient values obtained from the three experimental conditions described in Figure 3 and using the method reported earlier in Figure 2, which derives the diffusion coefficients (D) from the current steady-state region. For comparison, the diffusion coefficient (D) is also derived in the linear diffusion regime, which corresponds to region 1 in Figure 1 (short time) and is derived by the Cottrell equation.20 If Cm is assumed to be unchanged for a short period, then the diffusion current for short times of milliseconds can be solved:

(22)

Experimental Validation: Cobalt Deposition Description. Poretic polycarbonate membranes (overall diameter 13 mm) were used as templates. The average pore size (radius r0), density (N), and membrane thickness (L) are specified in Table 1. A 200-nm-thick gold layer serving as the back electrode is evaporated using a PVD process onto one side of the membrane. After evaporation, the membrane is fixed with the electrode facing down onto a conducting substrate (Cu plate) and a radius r of 0.5 cm is exposed to the electrolyte. Before mounting the membrane inside the cell, it is immersed in deionized water under ultrasonic agitation for a few minutes in order to get a good wetting of the pores and therefore a homogeneous growth over the surface area. Electroplating is done in a glass cell, with cryostat control of the temperature at 25 °C. The counter electrode is a Pt circular grid of about 10 cm2, and all of the experiments are performed under natural

I)

nFπr2(Cb - Cm)D1/2 π1/2t1/2

(23)

A clear dependence between pore sizes and diffusion coefficients is seen for both diffusion coefficient sets: that is, an increase of the diffusion coefficient is observed with increasing pore sizes. The pore-size-dependence of the diffusion coefficient has already been reported in the literature.15 The diffusion coefficient calculated in the linear diffusion regime is found to be up to 40 times higher than the one estimated for the spherical diffusioncontrolled regime. That is, the calculation of the diffusion coefficients in the current plateau region allows the estimation of the multidimensional kinetic process, which is representative of the effective nanowire growth regime. On the contrary, a Cottrell estimation of D does not represent the spherical mass transport limitation growth phenomena fully but rather a very short period of growth initiation where diffusion is mostly linear. Because the experimental diffusion coefficient values allow the

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J. Phys. Chem. C, Vol. 111, No. 13, 2007 5233

TABLE 2: Evaluation of the Global Diffusion Using Equation 12 by the Decomposition in Three Components Modeled Empirically by an Exponential Fitting experimental conditions

Dη (cm2s-1)

DL (cm2s-1)

Dr 0 (cm2s-1)

D (cm2s-1)

Cb ) 0.5 M/η ) -1 V/r0 ) 50 nm Cb ) 1 M/η ) -1 V/r0 ) 50 nm Cb ) 2 M/η ) -1 V/r0 ) 50 nm Cb ) 1 M/η ) -1.2 V/r0 ) 50 nm Cb ) 1 M/η ) -1.5 V/r0 ) 50 nm Cb ) 1 M/η ) -1 V/r0 ) 10 nm Cb ) 1 M/η ) -1 V/r0 ) 15 nm

2 × 10-6 2 × 10-6 2 × 10-6 5 × 10-6 2 × 10-5 2 × 10-6 2 × 10-6

6.5 × 10-7 1.5 × 10-6 7.0 × 10-6 1.5 × 10-6 1.5 × 10-6 1.5 × 10-6 1.5 × 10-6

1.8 × 10-6 1.8 × 10-6 1.8 × 10-6 1.8 × 10-6 1.8 × 10-6 1.8 × 10-7 2.4 × 10-7

1.95 × 10-6 2 × 10-6 3.8 × 10-6 3.4 × 10-6 6.8 × 10-6 1.73 × 10-6 1.74 × 10-6

prediction of concentration gradients and process kinetics, it is of great importance to evaluate D in the representative process regime region. As expressed in eq 14, we consider the diffusion coefficient dependence over three main parameters: pore radii, the concentration of metallic species in solution, and potentiostatic conditions. To deconvolute the three parameters during the steady-state current regime, D is evaluated by a series of experiments fixing two over the three main parameters. The exponential fitting of D for the variation of one parameter allowed us to derive a numerical expression of D for the case of cobalt deposition as follows:

Equation 26 derives the form of the pH difference for the case of the cobalt deposition:

E(Co2+/Co) ) -0.28 + 0.03 ln{[Co2+]b - [Co2+]m} E(H+/H2) ) 0.03 ln{10-2[pHb+pH ]} 0

η ) E(Co2+/Co) - E(H+/H2) ) -0.28 + 0.03 ln

D ) 1.4 × 10-4 exp[-2.3η]{9 × 10-14 exp[3152Cb] + 1 × 10-14 exp[1151292r0]}1/4 (24) Table 2 displays the values of D along with Dr0, DL, and Dη obtained by eq 14. The dominance of the electrostatic parameter on the overall process kinetics is clearly visible. With the deconvolution, it is possible to better discriminate the effect of each parameter on the global diffusional process. An analogue model was deduced to evaluate the duration of the electrodeposition as it is expressed in eq 25:

TP ) 19 exp[η]{44944 exp[-2.9Cb] +

(26)

|

ln

|pH0| )

{

{

(27)

}

[Co2+]b - [Co2+]m 10-2[pHb+pH ]

[Co2+]b - [Co2+]m

0

} |

exp[(η + 0.28/0.03)] + pHb 2 ln[10]

(28)

(29)

Figure 4b represents the pH difference value between the bulk pH and the one at the pore mouth as a function of the normalized time for the deposition inside three nanoelectrode pore sizes. At time 0, the pH difference is nil; it then increases (i.e., meaning that the pH at the pore mouth becomes bigger than the bulk pH) rapidly to reach a peak value and then decreases

324 exp[0.0348Veff]}1/4 (25) Concentration and pH Gradients Estimation. Figure 4a illustrates the concentration gradient between the pore mouth and the bulk solution obtained using eq 15 for different pore sizes with a bulk concentration of 1 M. The concentration gradients are drawn as a function of a normalized time (||T ||) in order to compare qualitatively the gradient on three different pore sizes. Time 0 represents the initiation of the growth process and time 1 represents the concentration gradient at the end of the growth process (TP). At short times and as soon as deposition is initiated, a concentration gradient is formed between the pore mouth and the bulk solution; the gradient decreases rapidly as the diffusion of metallic species at the pore mouth becomes multidimensional until it reaches a plateau. That would be maintained over the whole time-independent mass transport of the reaction. From very short times, there is an increasing concentration gradient with bigger pore size that is a higher diffusion coefficient D. This trend is expected as we measure a high diffusion coefficient at short and long times (Table 1) for bigger pore sizes, leading to a greater depletion of cobalt ions at the pore mouth. Because radial diffusion is proportionally greater to the other diffusional components in the steady-state current regime, we also expect a greater influence of the pore size over the concentration gradient in this region. This is seen clearly on the plot where a greater concentration gradient difference is predicted between the different pore sizes in the multidiffusionnal regime (time 0.8) than in the linear diffusion regime (time 0.1).

Figure 4. (a) Evaluation of the concentration gradient as a function of dimensionless time ||T || for each pore radius r0, at a bulk concentration of 1 M and an overvoltage of -1 V. (b) Evaluation of the pH difference pH0 as a function of ||T || for each pore radii r0, at a bulk concentration of 1 M and an overvoltage of -1 V.

5234 J. Phys. Chem. C, Vol. 111, No. 13, 2007

Philippe et al.

Figure 5. Simulation of the global diffusion D (eq 24) as a function of the bulk concentration Cb and the pore radius r0 for an overvoltage of -1 V.

Figure 6. Fitting of the cathodic Tafel equation (eq 30) in order to determine the transference number and the exchange current density of the reaction.

gradually until the end of the deposition process. A high pH inside the pores, compared to the bulk solution pH, is related to the high consumption of hydrogen species next to the anode. This leads to a higher concentration of hydroxyl species locally. Especially at short times, as soon as the electrode is polarized the reduction of protons occurs, leading to the considerable gradient of pH with the pores. As such, for a bulk pH of 3, we observe an increase of the pH near the anode leading to a local pH value of about 4.8. This gradient diminishes with deposition time as the nanopore length is decreasing. We observe a pH local increase that is less pronounced in smaller pore sizes, where diffusion of species inside the structure is slower in general and leads to reduced reaction kinetics. Model Prediction. Figure 5 displays the diffusion coefficient values simulation by using eq 24 for the case of cobalt deposition at overpotential of -1 V. For very big pore radii, the radial diffusion is no longer the limiting factor on process kinetics. For a pore radius of 100 nm, the concentration variation does not influence the diffusion coefficient values significantly. This proves that the linear diffusion reaches a constant when the pore radius increases. Transference Number, R, Estimation. At short time, a plot of current density versus t1/2 represents a straight line and the extrapolation to t ) 0 leads to the charge-transfer current density J(0). The values of J(0) obtained by extrapolation for all overvoltages are plotted versus |η| in Figure 6 for a nanoelectrode array of r0 50 nm and a cobalt concentration of 1 M. A straight line, the cathodic Tafel plot for the cobalt deposition, is obtained. Because the overvoltages used here are large (|η| . (RT/nF)), dissolution of the cobalt can be neglected. Fitting the cathodic Tafel equation (eq 30) for one reduction step, to the experimental data gives J(0) and the corresponding Rc of the reduction step, namely, Co2+ + 2e- f Co.

ln|J(0)| ) ln(2j0) +

(1 - Rc)F|η| RT

(30)

Figure 7. Evaluation of the heterogeneous electron-transfer rate as a function of the bulk concentration for each pore radius r0 using an overvoltage of -1 V.

The values obtained from this fit are j0 ) 3.7 mA‚cm-2 and Rc ) 0.85. They are reasonable and can be compared with data already reported in literature for cobalt.21 For redox reactions, the exchange current density would depend on the composition of the metal supporting an equilibrium reaction. The exchange current is also a complex function of the concentration of both the reactants and products involved in the specific reaction described by the exchange current. This function is particularly dependent on the shape of the charge-transfer barrier across the electrochemical interface. Heterogeneous Electron-Transfer Rate Estimation. Equation 31 manisfests the relationship between the dimentioneless parameter λ (eqs 8 and 10) and the kinetic regime

λ t

1/2

)

kf

(31)

D1/2

where kf is the heterogeneous rate constant for reduction of cobalt that is expressed by eq 32 as follows:

kf ) k0 exp

[-RRTF|η|]

(32)

Combining eqs 31 and 32 allows us to determine the heterogeneous rate constant of the reaction for different pore sizes of the nanoelectrode array:

k0 )

λ D1/2 -R F|η| t1/2 exp RT

[

]

(33)

The physical interpretation of the k0 is a measure of the kinetic facility of a redox couple. A system with a large k0 will achieve equilibrium on a short time scale, but a system with small k0 will be sluggish. Figure 7 displays the variation of k0 for three nanoelectrode pore sizes at three different concentrations and for an applied overvoltage of -1 V. In the whole range of concentration and pore sizes, we observe very low values of k0, indicating the kinetic irreversibility of the reaction toward deposition. For all concentrations, the trend shows the lowest value of k0 for a pore size of 100 nm. At lower pore size, we observe a linear increase of k0 that indicates logically the reduced facility of a deposition when the nanoelectrode becomes so small that diffusion of metallic species is becoming very low. For pore sizes bigger than 100 nm, we observe a slight increase of k0 until it reaches a plateau. The geometrical constraint leading to the variation in the kinetic facility of deposition is due to the relative contribution of the radial to the linear diffusion depending on the pore size. In other words, on the left part of the graph where pore sizes become smaller than 100 nm, the radial diffusion becomes the limiting factor to the deposition

Electrochemical Deposition of Metals kinetic. Around a r0 value of 100 nm, both radial and linear diffusion are of similar order and above this value the linear diffusion slowly becomes the limiting factor of deposition until radial diffusion can be considered as negligible for pore sizes bigger than 200 nm. The simulation of k0 also visualizes the kinetic facility of deposition depending on the cobalt concentration. For a concentration as low as 0.5 M, a lower k0 is logically simulated compared to more concentrated electrolytes because diffusion rates are smaller. Between electrolyte concentration of 1 M and 2 M, it is interesting to note a slight improvement of the deposition kinetic for 1 M. This is probably due to the fact that for a higher cobalt diffusion rate (2 M) the chargetransfer process becomes preponderant for the particular overvoltage condition used here. Conclusions An analytical expression for diffusion current is derived for the case of nanoelectrode arrays. The model takes into account the electrode geometry, the concentration gradient, and the fractional active area variation during the deposition. We show that the current form obtained provides an excellent quantitative and qualitative prediction of the current-time relationship. The kinetics of deposition is driven by a multidimensional diffusioncontrolled process that is varying as time elapses. We have decoupled three main parameters influencing the kinetics of the reaction: the nanoelectrode geometry, the concentration of metallic species in solution, and the potential regime. It clearly shows the interest in deriving a global diffusion coefficient in the steady-current state because the relative importance of each diffusional component is different than that from a short-time D derivation. The model validation was obtained using the deposition of cobalt. We illustrate the use of a kinetic model of the deposition to estimate concentration and pH gradients between the template surface and the nanoelectrode volume during the growth process. An analytical form of D and the duration process is derived and allows the predictions of the reaction kinetics depending on the synthesis parameters. Finally, the transference number and heterogeneous electron-transfer rate for cobalt deposition are also measured. A general kinetic model for electrochemical deposition inside nanoelectrodes has been developed successfully. The model can be used for describing quanlitatively and quantitatively a deposition process of any aqueous electrolyte inside any type of templated surface. This is an essential step in the building of a coherent correlation between electrochemical synthesis parameters, nanowire structures, and physical properties. Acknowledgment. Financial support by the Swiss State Secretariat for Education and Research in the frame of the European Project FP6-NMP MASMICRO is gratefully acknowledged. Abbreviations A r0

Fractional area (cm2) Pore radius (cm)

J. Phys. Chem. C, Vol. 111, No. 13, 2007 5235 Seq Veff r F L n N Cb Cm t TP D k0 kf R R T I J I(0) J(0) J0 λ

Equivalent surface area of the nanoelectrode (cm2) Effective volume (cm3) Radius of equivalent surface area (cm) Faraday’s constant (C) Membrane thickness (cm) Number of electrons transferred in a reaction Pore density (cm-2) Bulk concentration (M) Concentration at the pore mouth (M) Time (s) Process duration (s) Diffusion coefficient (cm2s-1) Standard heterogeneous rate constant (cms-1) Heterogeneous rate constant for reduction (cms-1) Transference number Gas constant (Jmol-1K-1) Temperature (K) Current (A) Current density (Acm-2) True exchange current (A) Pure charge-transfer current density (Acm-2) Exchange current density (Acm-2) Dimensionless kinetic parameter

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