J. Phys. Chem. C 2009, 113, 17215–17222
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Uptake of Molecular Species by Spherical Droplets and Particles Monitored Voltammetrically Francisco Martı´nez-Ortiz,† Eduardo Laborda,† Juan G. Limon-Petersen,‡ Emma I. Rogers,‡ ´ ngela Molina,*,† and Richard G. Compton‡ Carmen Serna,† Neil V. Rees,‡ A Departamento de Quı´mica Fı´sica, UniVersidad de Murcia, Espinardo 30100, Murcia, Spain, and Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: June 25, 2009; ReVised Manuscript ReceiVed: July 28, 2009
The uptake of molecular species by spherical particles is diverse in nature ranging from amalgamation processes at mercury droplets through ion transfer through the spherical interface between two immiscible electrolyte solutions (ITIES) (droplets, vesicles, etc.) to monitoring events in and around individual biological cells. Such processes can be controlled and monitored using voltammetry. We report the full characterization of such systems with reverse and differential pulse voltammetry. In particular, the former is a very useful diagnostic tool because of the appearance of a unique peak-shaped response in the reverse pulse voltammetry curves that arises solely as a consequence of the depletion of the species inside the particle. The greater this effect is, the smaller the size of the particle. 1. Introduction The assimilation of material by droplets can be promoted and studied voltammetrically. Processes at individual droplets and at arrays of these droplets have been investigated, and systems of interest range from oil droplets supported on electrode surfaces to individual mercury droplets themselves acting as electrodes.1-11 In the former example, the electrode potential can be used to induce uptake or release of species, while in the latter case, when the precursors of the material taken up are ions in solution, then the result of electrolysis is amalgam formation. Furthermore, Amatore has pioneered the use of voltammetry for monitoring processes in and around individual biological cells.12-14 In all of these examples, material enters or leaves the droplet, and transport processes within it are very important. In this paper, we build on previous insights15-20 into interfacial ion transfer and assimilation processes to develop the theory for two particularly insightful electrochemical techniques applied to the case where the product of an electrode reaction on the surface of a spherical electrode (droplet) diffuses inside the electrode (droplet), notably reverse pulse voltammetry (RPV) and differential pulse voltammetry (DPV). We show that both techniques are very well-suited to demonstrate that assimilation of the product by the electrode (or droplet) takes place. Moreover, their value for determining the diffusion coefficients of the chemical species inside and outside of the electrode (droplet) and the formal potential of the redox couple is demonstrated. In particular, RPV is very useful for the examination of these transport processes. Thus, the striking appearance of a unique peak-shaped response in the anodic branch of the RPV curve corresponding only to a microelectrode or microparticle is observed as a consequence of the depletion of the species inside. * Corresponding author. E-mail:
[email protected], phone: +34 868 88 7524, fax: +34 868 88 4148. † Universidad de Murcia. ‡ Oxford University.
This phenomenon is promoted by small particle size, long pulse duration, and large diffusion coefficient of the product (electrode phase) species. Experimental verification of the peak in RPV curves was performed with the Tl+/Tl(Hg) system at hemispherical mercury microelectrodes. Measurement of data for the diffusion coefficients of both species and the formal potential by means of RPV and DPV is also reported. 2. Experimental Section 2.1. Chemical Reagents. Thallium nitrate (TlNO3, Aldrich, 99.999%), ferrocene [Fe(C5H5)2, Aldrich, 98%], hexaammineruthenium(III) chloride ([Ru(NH3)6]3, Aldrich, 98%), acetonitrile (MeCN, Fischer Scientific, dried and distilled, 99%), tetra-nbutylammonium perchlorate (TBAP, Fluka, Puriss electrochemical grade, 99%), mercury(I) nitrate dihydrate [Hg2(NO3)2, Aldrich, >97%], potassium nitrate (KNO3, Aldrich, 99+% ACS reagent, 0.10 M), and nitric acid (HNO3, Fisher Scientific, 70%, 0.15 M) were all used as received without further purification. 2.2. Electrodes. Mercury hemispheres were electrodeposited on Pt disk electrodes of two different sizes, r0 ) 50 µm and r0 ) 25 µm, from a 10 mM solution of Hg2(NO3)2 with 0.1 M KNO3 as the supporting electrolyte (acidified with 0.5% of HNO3).21,22 Before deposition, the platinum electrode was calibrated by analyzing the steady-state voltammetry of a 2 mM solution of ferrocene in acetonitrile containing 0.1 M TBAP, assuming a value of diffusion coefficient of ferrocene in MeCN of D ) 2.3 × 10-5 cm2 s-1 at 25 °C.23 The mercury nitrate solution was bubbled with nitrogen gas for 30 min to remove atmospheric oxygen, and then mercury was deposited chronoamperometrically by holding the potential at -0.245 V, utilizing a coiled Pt wire as a counter electrode and an Ag wire as the reference electrode. The hemispherical geometry of the mercury deposit was established electrochemically by means of the ratio of the limiting steady-state currents of reduction of [Ru(NH3)6] 3+ from cyclic voltammetry before and after deposition, this ratio is close to π/2 (experimental ratio, 1.48 ( 0.03).
10.1021/jp905956t CCC: $40.75 2009 American Chemical Society Published on Web 09/09/2009
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Martı´nez-Ortiz et al. RPV and DPV experiments were performed with a homebuilt potentiostat, designed to apply the corresponding waveform (Figure 1a,c). A three-electrode setup was used, with a Hg hemispherical electrode as the working electrode, a Ag/AgCl reference electrode, and a coiled Pt wire as the counter electrode. Solutions were bubbled with N2 before experiments, and the inert atmosphere was assured. Before the application of each double pulse, the working electrode was open circuited for 10 s in order to restore equilibrium conditions. All experiments were undertaken at 298 ( 1 K in a heated Faraday cage. 3. Reverse Pulse Voltammetry Let us consider a reversible transfer reaction
O + ne- h R
Figure 1. Reverse pulse voltammetry: (a) Potential-time function and (b) RPV signal. Differential pulse voltammetry: (c) Potential-time function and (d) DPV signal.
2.3. Instrumentation. We used a computer-controlled µ-Autolab potentiostat type II (Eco-Chemie, Netherlands) to deposit the mercury as well as to calibrate the platinum disk and mercury hemispherical electrodes.
(1)
where O and R species are the oxidized and reduced species, or ionic species, at both sides of the interface. We assume that the reaction takes place at a stationary spherical electrode or particle, so that species R dissolves or transfers into the electrode (amalgamation) or particle. We consider the case of unequal diffusion coefficients DO * DR. According to the RPV waveform (Figure 1a), the first potential is a value corresponding to diffusion limiting conditions (E1 f -∞ when considering a reduction process) in the interval 0 e t e t 1 and changed from E1 to E2 for t g t1.
Figure 2. (a,b) RPV curves when species R diffuses in the electrode or particle for two radii: r0 ) 80 µm (a) and r0 ) 15 µm (b, numerical results).24 (c,d) Chronoamperometric curves corresponding to the second potential pulse (E2) for r0 ) 80 µm (c) and r0 ) 15 µm (d, numerical results).24 DO/DR ) 2, t1 ) 1 s, DO ) 10-5cm2 s-1, cO* ) 1 mM, and cR* ) 0 mM.
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J. Phys. Chem. C, Vol. 113, No. 39, 2009 17217 plane IRPV )
nFAc*O√DO
√π
(
1
-
√(t1 + t2)
γK2
√t2(1 + γK2)
)
(3)
Note that eq 2 is obtained under the assumption of semiinfinite diffusion inside the particle (so-called Koutecky´’s approximation, eq 6 in ref 24). This supposition should not be used, in general, for very small particles. Thus, we established by means of numerical calculations24 that this approximation is valid for ξ1 < 1, leading to an error smaller than 1.7%. So, as we will see in the following sections, the analytical equations are applicable until conditions at which RPV and DPV are suitable for the determination of the diffusion coefficients of both species and the formal potential. Cathodic Limiting Current: Id,DC (E2 f -∞, K2 f 0). It is fulfilled that Z2 f 0
[
p Id,DC ) Id,1 (t1 + t2) 1 +
√πDO(t1 + t2) r0
]
(4)
Under cathodic limiting conditions, the analytical equation is valid for any electrode or particle radius, including ultramicrosized ones (r0 f 0) and only depends on the species O. Anodic Limiting Current: Id,RPV (E2 f +∞, K2 f +∞). It is fulfilled that Y2 f 0, Z2 f -1, and Ω2 f -1/γ p p Id,RPV ) I1(t1 + t2) - Id,2 (t2)(1 + ξ2) - (γ + 1) · Id,2 (t2) · ξ1 ξ2 (1 + ξ2)[1 - F(ξ1)] + [G(R) - 1] + · S(ξ1, R) √π 2√π (5)
{
Figure 3. (a) Influence of the electrode or particle size on RPV curves when species R diffuses in the electrode or particle (solid line); DO/DR ) 2, t2 ) 0.05 s. (b) Influence of the DO/DR value on RPV curves when species R diffuses in a spherical electrode or particle with r0 ) 90 µm. Three values of DO/DR are considered: DO/DR ) 2 (hachured line), DO/DR ) 1 (solid line), and DO/DR ) 1/2 (dotted line). Other conditions are as in Figure 2.
By using a mathematical procedure based on Koutecky’s dimensionless parameter method, in a previous paper24 we obtained an expression for the current for a double-potential pulse for all values of E1 and E2 when the product R diffuses in the electrode. From this, the solution for RPV can be obtained by imposing the condition E1 f -∞ p IRPV ) I1(t1 + t2) + Id,2 (t2) · Z1,2 · (1 + γ · Ω2 · ξ2) -
{
p (t2) · Z1,2 (1 + γ · Ω2 · ξ2)[1 - F(ξ1)] + (γ + 1) · Id,2
ξ1
√π
[G(R) - 1] + γ · ξ2 · Y1,2 · Seven(R, ξ1) + γ · ξ2
√π
}
· Sodd(R, ξ1)
(2)
where the variables and functions are given in the Appendix considering K1 ) 0. From the above equation, a simpler one for planar particles is easily obtained by making r0 f ∞ [ξ1 f 0, F(ξ1) f 1 and ξ2 f 0]
}
Planar Electrodes.
plane p p Id,RPV ) Id,1 (t1 + t2) - Id,2 (t2) )
nFAc*O√DO
√πt2
(R - 1)
(6)
By comparing the equations for IRPV and Id,RPV with those for the case when species R is not taken up,25 we deduced that the RPV signal is sensitive to the assimilation process at spherical electrodes or particles, whereas at planar ones this does not depend on species R behavior as previously reported.26 Panels a and b of Figure 2 show RPV curves for two spherical particles of different sizes (r0 ) 80 µm and r0 ) 15 µm, respectively), when species R diffuses inside them. For each case, three t2 values are considered, showing the decrease of the Id,RPV/Id,DC value when t2 increases. The most interesting result is related to the shape of the RPV curve. With a large-sized particle, the “usual” form of RPV curves is obtained for any t2 value (Figure 2a), and with a smallsized particle (Figure 2b), a peak rather than a limiting value appears in the anodic branch of the RPV curve at large enough t2 values. To explain this anomalous behavior, we have plotted chronoamperometric curves corresponding to the second potential pulse for different E2 values in panels c and d of Figure 2. As shown in Figure 2c, for r0 ) 80 µm, the “usual” behavior of the I-t curves is encountered, so that the less positive the E2 value, the smaller the oxidation current at any time. In contrast, for smaller radii (Figure 2d), the decrease of the current under anodic limiting conditions (Id,RPV) is very fast as a consequence
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Figure 4. (a) Experimental (b) and analytical (s, eq 2) RPV curves at different t2 values corresponding to the reduction of Tl+ on a hemispherical mercury electrode with r0 ) 50 µm, t1 ) 300 ms, 0.2 mM TlNO3, and 0.1 M KNO3. (b) Experimental RPV curves at different t2 values corresponding to the reduction of Tl+ on a hemispherical mercury electrode with r0 ) 25 µm, t1 ) 800 ms, 0.2 mM TlNO3, and 0.1 M KNO3. (c) Experimental chronoamperometric curves of the second pulse for different E2 values under the conditions considered in Figure 4a. (d) Experimental chronoamperometric curves of the second pulse for different E2 values under the conditions considered in Figure 4b.
TABLE 1: Data of Diffusion Coefficients of Tl+(aq) and Tl(Hg) and the Formal Potential of the Redox Couple Obtained from RPV and DPV Experiments technique RPV DPV
medium
DTl(I) (cm2 s-1)
DTl(I)/DTl(Hg)
E0′ (mV)d
(aq) 0.1 M KNO3 (1.67 ( 0.03) × 10-5 a 2.05 ( 0.02a -394 (aq) 0.1 M KNO3 (1.70 ( 0.01) × 10-5 b 1.96 ( 0.03c -398
a Error bars calculated from the standard deviation of the values obtained from five values of Id,RPV and Id,DC. b Error bars calculated from the standard deviation of the best theoretical fits from five repeat potential step chronoamperograms. c Error bars calculated from the standard deviation of the values obtained from three repeat DPV peaks. d versus Ag/AgCl and KCl(sat).
of the depletion of the reduced species during the release step. Unlike what happens with medium-sized particles, this gives rise to the fact that the chronoamperometric curves cross at a given time (crossing time). After this time value, the Id,RPV is smaller than the current corresponding to a less anodic potential. This anomalous behavior is due to the the fact that at the less positive potential the depletion of species R is slower and so is the decay of the current with time. Accordingly, the appearance of the peak is promoted by those factors favoring the depletion of the species diffusing inside the electrode or particle, i.e., long pulse duration, small electrode or particle size, and large diffusion coefficient of species R. A similar behavior was previously depicted in normal pulse voltammetry, although in
this case, the reduced species must be initially present (amalgam electrodes) at the start of the experiment.27 Figure 3a shows the influence of the behavior of species R on the RPV signal. As deduced above from the analytical equations, under planar diffusion the response is the same regardless of whether species R diffuses in the electrode (or particle) or in the electrolytic solution, so that the solid and dotted curves are superimposed. However, at spherical electrodes or particles the uptake gives rise to an increase in the absolute value of anodic currents and to the shift of the null current potential toward more negative values; the greater both of the effects are, the smaller the electrode or particle radius. In Figure 3b, we have plotted RPV curves at a spherical particle with r0 ) 90 µm for three different ratios of diffusion coefficients [γ ) (DO/DR)1/2] when species R diffuses and is taken up. As is shown, the anodic limiting current is sensitive to the diffusion coefficient of species R, so that the value of Id,RPV increases with DR. Note that this tendency is contrary to that found when species R diffuses in the electrolytic solution.25 In order to validate the above theoretical results as well as to prove the feasibility of characterizing these systems with RPV, the experimental study of the reduction of Tl+ was performed on mercury hemispherical electrodes. In Figure 4b the experimental peak in the anodic branch of the RPV curve was obtained with a 25 µm radius mercury hemisphere. As stated above, the longer the electrode pulse is,
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Figure 6. Experimental (b) and analytical (s, eq 7) DPV curves corresponding to the reduction of Tl+ on a hemispherical mercury electrode with r0 ) 50 µm, t1 ) 400 ms, t2 ) 8 ms, 0.2 mM TlNO3, and 0.1 M KNO3. ∆E ) -50 mV (b), ∆E ) +50 mV (O).
us to precisely establish the formal potential. The values obtained are summarized in Table 1 and are in very good agreement with those existing in the literature (refs 22, 29 and references therein). It is worth highlighting the notable difference between the diffusion coefficients of the electroactive species, DO/DR, being around 2. This emphasizes the importance of having considered unequal diffusion coefficients for the electroactive species in our theoretical development because the common simplifying assumption DO ) DR may give rise to very significant errors. 4. Differential Pulse Voltammetry Figure 5. (a) Influence of the electrode or particle size on DPV curves when species R diffuses in the electrode or particle (solid line) and when it diffuses in the electrolytic solution (hachured line); DO/DR ) 2, t2 ) 0.02 s. (b) Influence of DO/DR value on RPV curves when species R diffuses in a spherical electrode/particle of r0 ) 90 µm. Three values of DO/DR are considered: DO/DR ) 2 (hachured line), DO/DR ) 1 (solid line), and DO/DR ) 1/2 (dotted line), t2 ) 0.02 s. Other conditions are as in Figure 2. Id,planar(t1) ) nFAc*O[(DO)1/2/(πt1)1/2].
the greater the depletion and therefore the greater the peak. This implies that there is a crossing of the chronoamperometric curves corresponding to different E2 values around the peak. This anomalous behavior of I-t curves is experimentally obtained and shown in Figure 4d. In Figure 4a, the characterization of the Tl+/Tl(Hg) couple is shown performed on a 50 µm radius mercury hemisphere. The experimental conditions such as electrode radius and pulse durations were selected so that the analytical solution obtained (eq 2) is valid, and the sensitivity of the techniques is high enough for the determination of both diffusion coefficients and the formal potential. Note that for the time values at which the RPV curves are recorded, the double-layer charging current is negligible compared to that due to the faradaic process of interest because the former rapidly decreases with time so that after 1-2 ms this is not significant.28 According to the approach proposed in a previous paper,25 the diffusion coefficient of the oxidized species (Tl+) is determined from the cathodic limiting current (eq 4), whereas the diffusion coefficient corresponding to the species diffusing inside the electrode (Tl0) is obtained from the anodic limiting one (eq 5). The location of the curve in the potential axis permits
In DPV (Figure 1c, d), the current is measured at the end of each pulse, and the difference between the two current samples [∆I ) I2(t1 + t2) - I1(t1)] is plotted versus a potential x-axis. The arithmetic average of both potential values [E1,2 ) (E1 + E2)/2] is selected as the x-axis potential.30 In DPV, the duration of the second pulse is often much shorter than the duration of the first pulse. According to this condition, a simple analytical expression for the DPV curve is obtained by making t2 , t1
{
p ∆I ) Id,2 · Z1,2 · (1 + γ · Ω2 · ξ2) 1 +
(γ + 1) [1 - F(ξ1′ )] (γ2K1 - 1)
}
(7)
where the variables and functions are defined in the Appendix. The validity of the approximation t2 , t1 was previously studied30 showing that for a ratio t1/t2 > 50 this involves nonsignificant errors for ξ1 < 1. Planar Electrodes (r0 f ∞). It is fulfilled that ξ1 f 0 [F(ξ1) f 1] and ξ2 f 0, so that eq 7 becomes p ∆Iplane ) Id,2 · Z1,2 )
nFAc*O√DO
√πt2
γ(K1 - K2) (1 + γK2)(1 + γK1)
(8)
As occurs in RPV, from the above equations, it is deduced that the DPV peak is affected by the behavior of the reduced species under spherical diffusion but insensitive under planar diffusion.
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TABLE 2: Parameters and Functions
2√DR(t1 + t2) r0J1
ξ1 )
√π · DR · t2
ξ2 )
R)
γ)
r0
( )
ηi )
t2 t1 + t2
DO DR
Ki ) exp(ηi), i ≡ 1, 2
Ji )
1 + γKi γ2Ki - 1
Y1,2 )
P1 )
sphericity of the particle corresponding to the duration of the second pulse
(A.3)
square root of the ratio of the duration of the second pulse and of the total time
(A.4)
square root of the ratio of the diffusion coefficients of the species outside and inside the particle
(A.5)
dimensionless potential
(A.7)
, i ≡ 1, 2
(A.8)
γ(K1 - K2) (1 + γK1)(1 + γK2)
(A.9)
K2(γ + 1)(γ · J2 - 1)J21 (1 + γK2)J22
(γ + 1) (γ2K1 - 1)
p Id,1 (t1 + t2) )
p Id,2 (t2) )
(A.2)
(A.6)
1 - Ki , i ≡ 1, 2 1 + γKi
Z1,2 )
sphericity of the particle corresponding to the total time
1/2
nF (Ei - E0 ), i ≡ 1, 2 RT
Ωi )
(A.1)
(A.11)
nFAc*O√DO
(A.12)
√π(t1 + t2)
nFAc*O√DO
(A.13)
√πt2 I1(t) )
(A.10)
p Id,1 (t) ·
(
1 1 + γK1
){
1+
√πDOt r0
[(
Figure 5a shows DPV curves when species R diffuses in the electrolytic solution and in the electrode or particle for three different radii. The response does not depend on the behavior
1 + γK1 1 - γ2K1
)
+
K1(γ + 1)2 (γ2K1 - 1)(1 + γK1)
]}
F(ξ1)
(A.14)
of species R under planar diffusion, but at spherical electrodes or particles the uptake process does affect the DPV signal. The smaller the electrode size is, the greater the effect. Thus, an
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J. Phys. Chem. C, Vol. 113, No. 39, 2009 17221 j-1
increase in peak current as well as a shift of the peak potential is observed where the product is soluble in solution and not the electrode (droplet or particle). Moreover, note that there is a significantly greater asymmetry between the DPV peaks corresponding to the normal (∆E < 0) and reverse (∆E > 0) mode when assimilation takes place. In Figure 5b, we have plotted the influence of γ ) (DO/DR)1/2 on the DPV curves when species R diffuses inside a spherical particle (r0 ) 90 µm). As is shown, ∆Ipeak increases with DR, unlike what is seen when the product species diffuses outside the particle.30 To verify the above theoretical results for DPV, we also performed an experimental study of the Tl+/Tl couple on a mercury hemispherical electrode with r0 ) 50 µm (Figure 6). As discussed above, at spherical electrodes the peak current depends on the ratio between the diffusion coefficients. Thus, the diffusion coefficient of the oxidized species was determined by single-step chronoamperometry and once this was known the DR value was obtained from the peak current. The formal potential was also determined from the location of the DPV curve on the potential axis. The values obtained from DPV compare well with those from RPV (Table 1) and with those in the literature (refs 22, 29 and references therein).
if 2j < i: A ) if 2j ) i: A )
2 · j!
l)0
2
G(x) ) x · arcsin(x) + √1 - x ∞
∑
i)2 j)1
{
(-1)j+i · (ξ1)i · R2j i
∏ l)1
pl
(A.16)
·A
}
]
(j - 1)! + l!(j - l - 1)!
()
j-1
∏ (i - 2l) l)0
if 2j > i, i odd: A ) i-3/2
∑ l)0
[ ()
×
2j · j!
J1 2l (j - 1)! · J2 l!(j - l - 1)! and i even: A ) 0
(-1)l ·
]
}
(A.18)
{
∑
(R, ξ1) )
S
i)1
j)0
(-1)j+i · (ξ1)i · R2j+1 · pi i
(i + 1)
·B
∏ pl l)1
}
(A.19)
being
j
if 2j + 1 < i: B )
[
∏ (i - 2l + 1) l)0
(2j + 1) · 2j · (j + 1)!
{
[]
j J1 1 (-1)m · 2m · + Y1,2 2 · γ · J1 J 2 m)0
∑
if 2j + 1 ) i: B ) j-1
∑
m)0
{
(-1)m · 2m ·
2m-1
× m
-k+1 ∏ ( j 2k -1 ) k)0
2Y1,2 1 × + (2j + 1) · γ · J1 (2j + 1)
[] J1 J2
2m-1
}
m
-k+1 ∏ ( j 2k -1 ) k)0 i
∏
}]
+
(-1)j(i + 1)j! pl J1 2j-1 Y1,2 l)1 P1 pi(2j + 1)! J2 where the sum is only effective forj g 1.
()
j
if 2j + 1 > i, i even: B )
2
·
J1 2(j-1) l)1 1 P1 J2 22jj! where the sum is only effective for j g 2.
(A.15)
with erfc being the complementary error function.
2l
2 j-1
i
∏ pl
(-1)j
Series.
Koutecký’s Function: F(x) ) e(x/2) · erfc(x/2)
J1 J2
(-1)l ·
∞
Acknowledgment. The authors greatly acknowledge Mr. Philip Hurst for the development of the potentiostat for the RPV and DPV experiments. A.M., F.M-O, C. S., and E.L. greatly appreciate the financial support provided by the Direccio´n General de Investigacio´n (MEC) (Project CTQ2006-12552/ BQU) and by the Fundacio´n SENECA (Expedient 03079/PI/ 05). Also, E.L. thanks the Ministerio de Ciencia e Innovacion for the grant received. E.I.R. and N.V.R. thank the EPSRC, and J.G.L-P thanks the CONACYT (Mexico) for funding.
J1 J2
1-
j
odd
In this paper, we have developed a theory to characterize the assimilation processes determined by the diffusion of the target species inside an electrode or particle, demonstrating that these processes are more easily detectable via electrochemistry when small electrodes are used. The study of differential pulse voltammetry, double-potential pulse chronoamperometry, and reverse pulse voltammetry was performed, showing that in the last case there exists a characteristic response associated with this kind of process. Moreover, DPV and RPV proved to be very valuable to determine the diffusion coefficients of the species under study inside and outside the particle.
being
[ ( )] ∑[ ()
l)0
j-2
5. Conclusions
Seven(R, ξ1) )
∏ (i - 2l)
[
1 + Y1,2 2 · γ · J1
i/2-1
∑
m)0
{
∏ (i - 2l + 1) l)0
(2j + 1) · 2j · (j + 1)!
(-1)m · 2m ·
[] J1 J2
2m-1
and i odd: B ) 0
m
×
-k+1 ∏ ( j 2k -1 ) k)0
}]
}
(A.20) (A.17)
References and Notes (1) Banks, C. E.; Davies, T. J.; Evans, R. G.; Hignett, G.; Wain, A. J.; Lawrence, N. S.; Wadhawan, J. D.; Marken, F.; Compton, R. G. Phys. Chem. Chem. Phys. 2003, 5, 4053. (2) Lledo-Fernande´z, C.; Hatay, I.; Ball, M. J.; Greenway, G. M.; Wadhawan, J. New J. Chem. 2009, 33, 749.
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(3) Rayner, D.; Fietkau, N.; Streeter, I.; Marken, F.; Buckley, B. R.; Page, P. C. B.; Campo, J.; Mas, R.; Mun˜oz, F. X.; Compton, R. G. J. Phys. Chem. C 2007, 111, 9992. (4) Evans, L. A.; Apreutesei, D.; Mehl, G. H.; Wadhawan, J. D. Electrochem. Commun. 2008, 10, 1720. (5) Horn, J. J.; Watson, A.; Lewis, M.; McCreedy, T.; Wadhawan, J. D. Electrochem. Commun. 2008, 10, 1384. (6) Schro¨der, U.; Compton, R. G.; Marken, F.; Bull, S. D.; Davies, S. G.; Gilmour, S. J. Phys. Chem. B 2001, 105, 1344. (7) Rassaei, L.; Sillanpa¨a¨, M.; Edler, K. J.; Marken, F. Electroanal. 2009, 21, 261. (8) Katif, N.; MacDonald, S. M.; Kelly, A. M.; Galbraith, E.; James, T. D.; Lubben, A. T.; Opallo, M.; Marken, F. Electroanal. 2008, 20, 469. (9) Niedziolka, J.; Rozniecka, E.; Stafiej, J.; Sirieix-Plenet, J.; Gaillon, L.; Caprio, D.; Opallo, M. Chem. Commun. 2005, 2954. (10) Mulder, W. H.; Sluyters, J. H. J. Electroanal. Chem. 1999, 468, 127. (11) Krujit, W. S.; Sluyters-Rehbach, M.; Sluyters, J. H. J. Electroanal. Chem. 1993, 351, 115. (12) Amatore, C.; Arbault, S.; Guille, M.; Lemaitre, F. Chem. ReV. 2008, 108, 2585. (13) Amatore, C.; Arbault, S.; Bonifas, I.; Guille, M.; Lemaitre, F.; Verchier, Y. Biohys. Chem. 2007, 129, 181. (14) Amatore, C.; Oleinick, A.; Svir, I. J. Electroanal. Chem. 2005, 575, 103. (15) Koryta, J. Electrochim. Acta 1979, 24, 293. (16) Kakiuchi, T. Electrochim. Acta 1998, 44, 171. (17) Reymond, F.; Fermı´n, D.; Lee, H. J.; Girault, H. H. Electrochim. Acta 2000, 45, 2647.
Martı´nez-Ortiz et al. (18) Ortun˜o, J. A.; Serna, C.; Molina, A.; Gil, A. Anal. Chem. 2006, 78, 8129. (19) Molina, A.; Serna, C.; Ortun˜o, J. A.; Gonza´lez, J.; Torralba, E.; Gil, A. Anal. Chem. 2009, 81, 4220. (20) Molina, A.; Serna, C.; Gonza´lez, J.; Ortun˜o, J. A.; Torralba, Phys. Chem. Chem. Phys. 2009, 11, 1159. (21) Mauzeroll, J.; Hueske, E. A.; Bard, A. J. Anal. Chem. 2003, 75, 3880. (22) Limon-Petersen, J. G.; Streeter, I.; Rees, N. V.; Compton, R. G. J. Phys. Chem. C 2008, 112, 17175. (23) Marsh, K. N.; Deev, A.; Wu, A.C.-T.; Tran, E.; Klamt, A. Kor. J. Chem. Eng. 2002, 19, 357. (24) Molina, A.; Compton, R. G.; Serna, C.; Martı´nez-Ortiz, F.; Laborda, E. Electrochim. Acta 2009, 54, 2320. (25) Laborda, E.; Rogers, E. I.; Martı´nez-Ortiz, F.; Limon-Petersen, J. G. Rees, N. V.; Molina, A.; Compton, R. G. J. Electroanal. Chem. 2009, 634, 1. (26) Heyrovsky´, J.; Kuta, J. Principles of Polarography; Academic Press: New York, 1966; Chapter XVII. (27) Molina, A.; Serna, C.; Martı´nez-Ortiz, F.; Laborda, E. J. Electroanal. Chem. 2008, 617, 14. (28) Compton, R. G.; Banks, C. E. Understanding Voltammetry; World Scientific: Hackensack, NJ, 2007; Chapter 9. (29) Limon-Petersen, J. G.; Rees, N. V.; Streeter, I.; Molina, A.; Compton, R. G. J. Electroanal. Chem. 2008, 623, 165. (30) Molina, A.; Laborda, E.; Rogers, E. I.; Martı´nez-Ortiz, F.; Serna, C.; Limon-Petersen, J. G.; Rees, N. V.; Compton, R. G. J. Electroanal. Chem. 2009, 634, 73.
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