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Corrigendum to “Voltammetric studies of colloidal particle monolayer on a gold rotating disk electrode” [Colloids Surf. A: Physicochem. Eng. Aspects 403 (2012) 62–68]
Colloids and Surfaces A: Physicochem. Eng. Aspects 429 (2013) 159–161
Contents lists available at SciVerse ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Corrigendum
Corrigendum to “Voltammetric studies of colloidal particle monolayer on a gold rotating disk electrode” [Colloids Surf. A: Physicochem. Eng. Aspects 403 (2012) 62–68] a ´ Magdalena Nosek a,∗ , Paweł Weronski , Paweł Nowak a , Jakub Barbasz a,b a b
Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30-239 Krakow, Poland Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 20 March 2013 Accepted 27 March 2013 Available online 14 May 2013
The authors regret that the derivation of Eq. (7) in Ref. [1], describing the limiting diffusion current at a rotating disk electrode with a monolayer of spherical particles, is incorrect. The starting point of the derivation was the equation for the limiting diffusion current I at a rotating disk electrode covered with a gel membrane [2] of the thickness ım
I = IL 1 +
DCım Dm Cm ıd
−1
(1)
where D and Dm are the diffusivities of reactant in solution and in the membrane, respectively, C and Cm are the solute concentrations at the membrane-solution interface in solution and in the membrane, respectively, IL is the limiting diffusion current in the absence of the membrane, and ıd is the thickness of the diffusion boundary layer. Assuming negligible convection in the layer of adsorbed particles, we used Eq. (1) to find the limiting diffusion current at a rotating disk electrode with particles. For that, we replaced the parameters Cm and Dm , appearing in Eq. (1), with the corresponding parameters in a porous layer of the same thickness: Cm = εC
(2)
the porous layer. The symbols ε and , appearing in Eqs. (2) and (3), denote the average porosity and tortuosity of the porous layer, respectively. As a consequence we obtained the equation
I = IL 1 +
Dm
(3)
i.e., with the superficial solute concentration in the porous layer at the layer-solution interface and with the effective diffusivity in
−1
,
(4)
which, after substituting the formulae for the average tortuosity and porosity [1], gave the before-mentioned Eq. (7). It should be noticed, however, that the definition of effective diffusivity in a porous medium has been introduced to describe diffusion through the pore space of porous media in contrary to its superficial volume. Therefore, we should replace Cm with the solution concentration C instead of the superficial concentration εC. Unfortunately, by substituting Eqs. (2) and (3) into Eq. (1) we obtained in Eq. (4) the porosity in the second power, resulting in an overestimation of the effect of porosity on the limiting diffusion current. To make the error more obvious, we can derive Eq. (7) of Ref. [1] in the correct form by considering the diffusion permeability of a particle layer. As demonstrated in Ref. [2], the limiting diffusion current at a rotating disk electrode covered with a layer of gel is equal, in terms of diffusion permeability,
and Dε = ,
ım ε2 ıd
I = IL 1 +
Ps Pm
−1 ,
(5)
where Pm =
D˛ −j = c ım
(6)
and DOI of original article: http://dx.doi.org/10.1016/j.colsurfa.2012.03.056. ∗ Corresponding author. Tel.: +48 12 6395 126. E-mail address: [email protected] (M. Nosek). 0927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.colsurfa.2013.03.057
Ps =
D ıd
(7)
160
M. Nosek et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 429 (2013) 159–161
Fig. 1. Schematic representation of porous layer adsorbed on electrode surface.
are, respectively, the diffusion permeability of the membrane and solution diffusion layer. The symbols j and ˛ appearing in Eq. (6) denote the superficial diffusion flux and membrane partition coefficient, respectively. We can use Eq. (5) to calculate the limiting diffusion current at a rotating disk electrode covered with a particle monolayer, as long as the convection in the layer is negligible. For that, we need to find the superficial diffusion permeability of particle monolayer. Let us consider a porous layer adsorbed on the surface of electrode, assuming a constant porosity and tortuosity of the layer (see Fig. 1), as well as a pore size much larger than that of the solute molecule. Considering that the reactant concentration Cl in the liquid phase is a continuous function of the distance z, we can express the stationary diffusion flux of reactant in the liquid phase of the layer by the equation jl = −
1 dn C (ı) − Cl (0) , = −D l Sm dt ım
(8)
where Sm = ε S is the surface area of the liquid phase cross-section at a given distance z, S is the surface area of electrode, and n is the number of moles of solute molecules crossing the surface Sm in the time t. Substituting Cl (ım ) = C and Cl (0) = 0 we get the superficial diffusion flux in the porous layer j=−
1 dn Dε =− C, S dt ım
(9)
and so Pm =
−j Dε . = C ım
(10)
After substituting Eqs. (7) and (10) into Eq. (5) we obtain
I = IL 1 +
ım εıd
−1 ,
(11)
where, in contrary to Eq. (4), the porosity appears in the first power. In the case of particle layer we can replace ım with 2r, where r is the particle radius. Then, using the equations for the average porosity and tortuosity [1], we get Eq. (7) of Ref. [1] in the correct form:
I = IL
2r[1 + (/8 − 1/3)] 1+ ıd (1 − 2/3)
−1
.
(12)
Fig. 2. Limiting diffusion current as a function of number of rotations per minute. Solid line presents theoretical predictions for bare electrode. Dashed and dotted lines denote theoretical predictions for electrode with particle layer at coverage 0.373 and particle radius 1.51 m using Eq. (12) and Eq. (7) of Ref. [1], respectively. Open circles denote experimental results.
Quantitatively, the effect of the error on the calculated diffusion current depends on the average porosity and the ratio of thickness of solution diffusion layer to that of particle layer:
1 ∂I I ≈ ε = I I ∂ε
1+
εıd 2r
−1
ε , ε
(13)
where I and ε denote the current and porosity determination errors, respectively. The effect of tortuosity is minor because its average value is close to unity. For the values reported in Ref. [1], i.e., = 0.3 and r = 0.5 m, at the highest number of rotations of the disk ıd = 5.5 m and we get ε I ≈ 0.19 . I ε
(14)
The relative porosity error, resulting from taking the porosity in the second power, is equal
ε 0.82 − 0.8 = = 0.2, ε 0.8
(15)
thus giving the maximum relative error of diffusion current about 3.8%. This estimation suggests that the current determination errors in our experiments, resulting from the incorrect formula, were small and comparable to the errors in other variables, such as the surface coverage or particle diameter. Therefore, the agreement between our theoretical and experimental results was fine and we overlooked the error in the derivation of Eq. (7) in Ref. [1]. To verify Eq. (12) experimentally we have changed the parameters of our system in such a way to make the diffusion current more sensitive to variations of porosity. Thus, for the coverage 0.373, particle radius 1.51 m, and diffusion layer thickness ıd = 10.9 m the constant of proportionality between the relative errors of current and porosity is 0.37, the relative porosity error equals about 0.25, and the relative current error is greater than 9%. This value is large enough to discriminate between the correct and incorrect models. In Fig. 2 we have presented a comparison of our experimental results and results predicted by the two theoretical models. As we can see, the experimental results agree much better with Eq. (12) than with Eq. (7) of Ref. [1], especially at the large number of rotations per minute of the rotating disk electrode, where deviations of diffusion current predicted by the incorrect model increase.
M. Nosek et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 429 (2013) 159–161
Finally, let us make a remark on the accuracy of the theoretical model at high surface coverages. Because of the basic assumption of constant porosity of the porous layer, we may expect a decrease of accuracy of the model in the range of very high surface coverage, where the assumption is not well satisfied. The variation of porosity with the distance from the adsorption surface can be estimated by the difference between its greater value ε = 1 at z = 1 and its average value equal 1 − 2/3. Thus, the porosity variation is of the order of 2/3. While at the coverage 0.3 the variation is about 0.2 and the average porosity is 0.8, at the coverage 0.9 (close to the hexagonal packing of hard spheres) the variation achieves 0.6 at the average porosity 0.4, meaning the variations of porosity are larger than its average value. In this range of coverage we can expect poorer agreement between the model and experiment. Indeed, our recent investigations suggest the inaccuracy in diffusion current calculated with the model for 1600 rpm, resulting from neglecting
161
variations of porosity in the layer of spherical particles, at the coverage 0.3, 0.6, and 0.9 is equal 0.28%, 2.6%, and 27%, respectively. The discussion and conclusions of the paper are not affected by the error in the derivation of Eq. (7). The authors apologize for the error. Acknowledgment We thank MSc. Eng. Piotr Batys for help in preparation of the manuscript. References ´ [1] M. Nosek, P. Weronski, P. Nowak, J. Barbasz, Colloids Surf. A: Physicochem. Eng. Aspects 403 (2012) 62–68. [2] D.A. Gough, J.K. Leypoldt, Anal. Chem. 51 (3) (1979) 439–444.
Contents lists available at SciVerse ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Corrigendum
Corrigendum to “Voltammetric studies of colloidal particle monolayer on a gold rotating disk electrode” [Colloids Surf. A: Physicochem. Eng. Aspects 403 (2012) 62–68] a ´ Magdalena Nosek a,∗ , Paweł Weronski , Paweł Nowak a , Jakub Barbasz a,b a b
Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30-239 Krakow, Poland Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 20 March 2013 Accepted 27 March 2013 Available online 14 May 2013
The authors regret that the derivation of Eq. (7) in Ref. [1], describing the limiting diffusion current at a rotating disk electrode with a monolayer of spherical particles, is incorrect. The starting point of the derivation was the equation for the limiting diffusion current I at a rotating disk electrode covered with a gel membrane [2] of the thickness ım
I = IL 1 +
DCım Dm Cm ıd
−1
(1)
where D and Dm are the diffusivities of reactant in solution and in the membrane, respectively, C and Cm are the solute concentrations at the membrane-solution interface in solution and in the membrane, respectively, IL is the limiting diffusion current in the absence of the membrane, and ıd is the thickness of the diffusion boundary layer. Assuming negligible convection in the layer of adsorbed particles, we used Eq. (1) to find the limiting diffusion current at a rotating disk electrode with particles. For that, we replaced the parameters Cm and Dm , appearing in Eq. (1), with the corresponding parameters in a porous layer of the same thickness: Cm = εC
(2)
the porous layer. The symbols ε and , appearing in Eqs. (2) and (3), denote the average porosity and tortuosity of the porous layer, respectively. As a consequence we obtained the equation
I = IL 1 +
Dm
(3)
i.e., with the superficial solute concentration in the porous layer at the layer-solution interface and with the effective diffusivity in
−1
,
(4)
which, after substituting the formulae for the average tortuosity and porosity [1], gave the before-mentioned Eq. (7). It should be noticed, however, that the definition of effective diffusivity in a porous medium has been introduced to describe diffusion through the pore space of porous media in contrary to its superficial volume. Therefore, we should replace Cm with the solution concentration C instead of the superficial concentration εC. Unfortunately, by substituting Eqs. (2) and (3) into Eq. (1) we obtained in Eq. (4) the porosity in the second power, resulting in an overestimation of the effect of porosity on the limiting diffusion current. To make the error more obvious, we can derive Eq. (7) of Ref. [1] in the correct form by considering the diffusion permeability of a particle layer. As demonstrated in Ref. [2], the limiting diffusion current at a rotating disk electrode covered with a layer of gel is equal, in terms of diffusion permeability,
and Dε = ,
ım ε2 ıd
I = IL 1 +
Ps Pm
−1 ,
(5)
where Pm =
D˛ −j = c ım
(6)
and DOI of original article: http://dx.doi.org/10.1016/j.colsurfa.2012.03.056. ∗ Corresponding author. Tel.: +48 12 6395 126. E-mail address: [email protected] (M. Nosek). 0927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.colsurfa.2013.03.057
Ps =
D ıd
(7)
160
M. Nosek et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 429 (2013) 159–161
Fig. 1. Schematic representation of porous layer adsorbed on electrode surface.
are, respectively, the diffusion permeability of the membrane and solution diffusion layer. The symbols j and ˛ appearing in Eq. (6) denote the superficial diffusion flux and membrane partition coefficient, respectively. We can use Eq. (5) to calculate the limiting diffusion current at a rotating disk electrode covered with a particle monolayer, as long as the convection in the layer is negligible. For that, we need to find the superficial diffusion permeability of particle monolayer. Let us consider a porous layer adsorbed on the surface of electrode, assuming a constant porosity and tortuosity of the layer (see Fig. 1), as well as a pore size much larger than that of the solute molecule. Considering that the reactant concentration Cl in the liquid phase is a continuous function of the distance z, we can express the stationary diffusion flux of reactant in the liquid phase of the layer by the equation jl = −
1 dn C (ı) − Cl (0) , = −D l Sm dt ım
(8)
where Sm = ε S is the surface area of the liquid phase cross-section at a given distance z, S is the surface area of electrode, and n is the number of moles of solute molecules crossing the surface Sm in the time t. Substituting Cl (ım ) = C and Cl (0) = 0 we get the superficial diffusion flux in the porous layer j=−
1 dn Dε =− C, S dt ım
(9)
and so Pm =
−j Dε . = C ım
(10)
After substituting Eqs. (7) and (10) into Eq. (5) we obtain
I = IL 1 +
ım εıd
−1 ,
(11)
where, in contrary to Eq. (4), the porosity appears in the first power. In the case of particle layer we can replace ım with 2r, where r is the particle radius. Then, using the equations for the average porosity and tortuosity [1], we get Eq. (7) of Ref. [1] in the correct form:
I = IL
2r[1 + (/8 − 1/3)] 1+ ıd (1 − 2/3)
−1
.
(12)
Fig. 2. Limiting diffusion current as a function of number of rotations per minute. Solid line presents theoretical predictions for bare electrode. Dashed and dotted lines denote theoretical predictions for electrode with particle layer at coverage 0.373 and particle radius 1.51 m using Eq. (12) and Eq. (7) of Ref. [1], respectively. Open circles denote experimental results.
Quantitatively, the effect of the error on the calculated diffusion current depends on the average porosity and the ratio of thickness of solution diffusion layer to that of particle layer:
1 ∂I I ≈ ε = I I ∂ε
1+
εıd 2r
−1
ε , ε
(13)
where I and ε denote the current and porosity determination errors, respectively. The effect of tortuosity is minor because its average value is close to unity. For the values reported in Ref. [1], i.e., = 0.3 and r = 0.5 m, at the highest number of rotations of the disk ıd = 5.5 m and we get ε I ≈ 0.19 . I ε
(14)
The relative porosity error, resulting from taking the porosity in the second power, is equal
ε 0.82 − 0.8 = = 0.2, ε 0.8
(15)
thus giving the maximum relative error of diffusion current about 3.8%. This estimation suggests that the current determination errors in our experiments, resulting from the incorrect formula, were small and comparable to the errors in other variables, such as the surface coverage or particle diameter. Therefore, the agreement between our theoretical and experimental results was fine and we overlooked the error in the derivation of Eq. (7) in Ref. [1]. To verify Eq. (12) experimentally we have changed the parameters of our system in such a way to make the diffusion current more sensitive to variations of porosity. Thus, for the coverage 0.373, particle radius 1.51 m, and diffusion layer thickness ıd = 10.9 m the constant of proportionality between the relative errors of current and porosity is 0.37, the relative porosity error equals about 0.25, and the relative current error is greater than 9%. This value is large enough to discriminate between the correct and incorrect models. In Fig. 2 we have presented a comparison of our experimental results and results predicted by the two theoretical models. As we can see, the experimental results agree much better with Eq. (12) than with Eq. (7) of Ref. [1], especially at the large number of rotations per minute of the rotating disk electrode, where deviations of diffusion current predicted by the incorrect model increase.
M. Nosek et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 429 (2013) 159–161
Finally, let us make a remark on the accuracy of the theoretical model at high surface coverages. Because of the basic assumption of constant porosity of the porous layer, we may expect a decrease of accuracy of the model in the range of very high surface coverage, where the assumption is not well satisfied. The variation of porosity with the distance from the adsorption surface can be estimated by the difference between its greater value ε = 1 at z = 1 and its average value equal 1 − 2/3. Thus, the porosity variation is of the order of 2/3. While at the coverage 0.3 the variation is about 0.2 and the average porosity is 0.8, at the coverage 0.9 (close to the hexagonal packing of hard spheres) the variation achieves 0.6 at the average porosity 0.4, meaning the variations of porosity are larger than its average value. In this range of coverage we can expect poorer agreement between the model and experiment. Indeed, our recent investigations suggest the inaccuracy in diffusion current calculated with the model for 1600 rpm, resulting from neglecting
161
variations of porosity in the layer of spherical particles, at the coverage 0.3, 0.6, and 0.9 is equal 0.28%, 2.6%, and 27%, respectively. The discussion and conclusions of the paper are not affected by the error in the derivation of Eq. (7). The authors apologize for the error. Acknowledgment We thank MSc. Eng. Piotr Batys for help in preparation of the manuscript. References ´ [1] M. Nosek, P. Weronski, P. Nowak, J. Barbasz, Colloids Surf. A: Physicochem. Eng. Aspects 403 (2012) 62–68. [2] D.A. Gough, J.K. Leypoldt, Anal. Chem. 51 (3) (1979) 439–444.