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Cyclic and Square-Wave Voltammetry at Diffusionally Asymmetric Microscopic and Nanoscopic Liquid−Liquid Interfaces: A Simple Theoretical Approach A. Molina,*,† E. Laborda,† and R. G. Compton*,‡ †

Departamento de Química Física, Facultad de Química, Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia, 30100 Murcia, Spain ‡ Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom ABSTRACT: The theoretical modeling of reversible ion transfers across liquid−liquid interfaces supported at micro/nanocapillaries is developed by means of a simple, approximate analytical approach. A very simple analytical solution is obtained for any multipulse technique and applied to the case of the two most widely used voltammetric techniques: cyclic and square-wave voltammetries. The analytical results fully describe the shape and position of the voltammograms as well as the effect of the key variables of the system: capillary size, time scale of the experiment, and ion diffusion coefficients and bulk concentrations. Their influence is analyzed and quantified with simple equations that immediately enable the determination of the ion-transfer formal potential.

and pores as well as arrays of them have been employed6 since the pioneering work by Girault et al.7 Despite the practical benefits of the use of micro- and nanointerfaces, the theoretical treatment is more complex with respect to macrointerfaces, the 1-D problem turning 2-D. This has usually been tackled by means of numerical methods,8−11 which can be excessively time-consuming when dealing with very small interfaces and require computer programming knowledge. A simple theoretical framework is developed in this work based on an approximate analytical resolution of the diffusionally

1. INTRODUCTION Monitoring and control of ion transfer across liquid−liquid interfaces by means of electrochemical methods have attracted much attention given the fundamental and practical interest of such processes.1 For instance, these are ubiquitous in energy conversion processes in nature 2 and under study for technological ones;3 they are also excellent model systems to gain further insight into the uptake and release of drugs and other compounds by biological structures.2,4,5 In the last three decades, one of the main advances in this field has been the design of platforms for the reduction of the interface size to minimize distorting effects (such as Ohmic drop), miniaturize devices, and study the kinetics of very fast transfer processes. Thus, micrometric and nanometric pipets © 2014 American Chemical Society

Received: June 5, 2014 Revised: July 9, 2014 Published: July 9, 2014 18249

dx.doi.org/10.1021/jp505593f | J. Phys. Chem. C 2014, 118, 18249−18256

The Journal of Physical Chemistry C

Article

differences between the ion diffusivities in each phase. The influence on the response of the key system variables as well as of others associated with the electrochemical techniques is examined, pointing out important differences with respect to the behavior reported at macrointerfaces. Finally, simple expressions are given for the determination of the transfer formal potential from the half-wave potential of the ion ingress sigmoid and the ion egress peak potential in CV and from the forward peak potential in SWV.

2. THEORY Consider the system described in Figure 1 corresponding to the transfer of the ion XΖ across a liquid−liquid interface supported at a nanocapillary under external polarization. The resulting mass transport of the target ion under fully supported conditions takes place by diffusion, and then it can be described by Fick’s second law α ⎡ ∂ 2c α ∂c Xα ∂ 2c Xα 1 ∂c X ⎤ ⎥ = DXα ⎢ 2X + + ∂t r ∂r ⎦ ∂r 2 ⎣ ∂z

(1)

The superscript α refers to the phase (inner, α = in; outer, α = out) where the ion XΖ diffuses, and DαX is the diffusion coefficient of the target ion. Considering that the transfer is reversible and a constant potential pulse is applied, the boundary value problem is given by ⎫ out ⎬c X = c X*,out t > 0, z → +∞⎭ t = 0, z ≥ 0

⎫ in ⎬c X = c X*,in t > 0, z → −∞ , 0 ≤ r ≤ a ⎭ t = 0, z ≤ 0, 0 ≤ r ≤ a t > 0, z = 0, 0 ≤ r ≤ a ⎛ ∂c in ⎞ ⎛ ∂c out ⎞ = DXout ⎜ X ⎟ DXin⎜ X ⎟ ⎝ ∂r ⎠ z = 0 ⎝ ∂r ⎠ z = 0 Figure 1. Diffusion fields (obtained as described in Section 2.2) corresponding to an ion transfer across a liquid−liquid interface supported on a micro/nanocapillary of radius a.

c Xs,out = e ηc Xs,in

(2)

whereas a is the radius of the capillary, cX* and are the bulk and surface concentrations of species XΖ, respectively, and η is the dimensionless overpotential ,in/out

asymmetric problem corresponding to the ion transfer at micro/ nanocapillaries. (See Figure 1.) Whereas the ion mass transport at the inner side of the capillary is governed by quasi-linear diffusion, convergent diffusion is very significant in the outer solution. Thus, within this approximate treatment, the ion flux toward/from the inner interface will be assumed to have a form analogous to that at macrointerfaces,12,13 whereas that at the outer interface is analogous to that obtained for disc microelectrodes under steady-state conditions.14,15 Under these simplifying hypotheses, a very simple closed-form expression is obtained for the current response of reversible ion transfers valid for any multipulse voltammetric technique, and it is applied to the most widely used methods in the electrochemical study of ion-transfer processes: cyclic voltammetry (CV)16 and squarewave voltammetry (SWV).17 From the comparison with rigorous numerical results, the analytical solutions are found to fully describe the shape and position of the CV and SWV signals as well as its variation when altering the time scale of the experiments and the radius of the capillary. This suggests that despite the simplicity of the mathematical treatment the solution accounts for the asymmetry of the diffusion fields as well as possible large

η=

ΖF in in 0 ′ (Δout ϕ − Δout ϕ ) RT

cs,in/out X

(3)

Δinoutϕ

where Ζ is the charge of the target ion, is the electric potential difference across the interface, and Δinoutϕ0′ is the formal potential of the ion transfer, and the other symbols have their usual meanings. 2.1. Analytical Treatment for Multipulse Techniques. Provided that the diameter of the capillary is on the (sub)micrometric scale, the diffusion of the ion in the inner solution can be modeled as linear in an approximate way, and steady-state conditions can be assumed in the outer solution. Applying the interfacial flux conservation principle and after adequate mathematical manipulations, the above considerations have led to accurate analytical expressions for single- and double-potential pulse techniques.18 However, the modeling of multipulse techniques (such as CV and SWV) is more challenging due to the accumulative effects associated with the “memory” of the system of previous potential pulses, which can compromise the applicability of analytical mathematical methods. Here we address this issue by considering the 18250



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voltammetric technique. As previously indicated, linear diffusion is assumed to prevail in the inner phase in an approximate way ⎛ ∂ 2c in(n) ⎞ ∂c Xin(n) ≈ DXin⎜⎜ X2 ⎟⎟ ∂tn ⎝ ∂z ⎠

(4)

and steady-state conditions in the outer phase out(n) ⎤ ⎡ ∂ 2c out(n) ∂c Xout(n) ∂ 2c Xout(n) 1 ∂c X ⎥ = 0 = DXout⎢ X 2 + + 2 ∂tn r ∂r ⎥⎦ ∂r ⎣⎢ ∂z

(5)

Attending to the linearity of the diffusion operator in eq 4, the solution of the inner phase can be written as the linear combination c Xin(n)(z , t ) = c Xin(n − 1)(z , t ) + c X̃ in(n)(z , tn)

(6)

where t corresponds to the time elapsed from the beginning of the experiment (i.e., the application of the first pulse) and cin(n−1) (z,t) is the solution obtained for the previous pulse, X n − 1. Thus, with regard to the ion concentration on the inner solution, the problem now corresponds to the determination of the partial solution c̃in(n) X (z,tn) with null limit and initial conditions ⎫ ⎬c X̃ in(n) = 0 tn > 0, z → −∞ , 0 ≤ r ≤ a ⎭ tn = 0, z ≤ 0, 0 ≤ r ≤ a

(7)

Also, the boundary value problem establishes that the flux conservation (eq 8) and Nernstian equilibrium (eq 9) hold at the liquid−liquid interface ⎧⎛ in(n − 1) ⎞ ⎛ ∂c ̃ in(n) ⎞ ⎫ ⎪ ∂c ⎪ ⎟⎟ πa 2DXin ⎨⎜⎜ X + ⎜⎜ X ⎟⎟ ⎬ ⎪⎝ ∂z ⎠ ⎝ ∂z ⎠ z = 0 ⎪ ⎩ ⎭ z=0 = −2πDXout

∫0

a

⎛ ∂c out(n) ⎞ ⎜⎜ X ⎟⎟ ⎝ ∂z ⎠

r dr z=0

c Xs,out(n) = e ηnc Xs,in(n)

(8) (9)

where 2πDXout

∫0

a

⎛ ∂c out(n) ⎞ ⎜⎜ X ⎟⎟ ⎝ ∂z ⎠

r dr = 4DXouta(c X*,out − c Xs,out(n)) z=0

⎛ ∂c ̃ in(n) ⎞ c ̃ s,in(n) = − πa2DXin X πa2DXin⎜⎜ X ⎟⎟ ⎝ ∂z ⎠z = 0 πDXintn (10)

According to the above definitions and by applying conditions 7−9, the following expressions are obtained for the interface concentrations of the ion ⎛ c *in + ξ c *out ⎞ n ⎛ 1 + ξje ηj−1 ⎞ 1 X ⎟ c Xs,in(n) = ⎜ X ⎟ ∏ ⎜⎜ η η ⎟ ⎝ 1 + ξ1e 1 ⎠ j = 2 ⎝ 1 + ξje j ⎠ c Xs,out(n)

Figure 2. Comparison of the analytical (eq 13) and numerical (Section 2.2) solutions in cyclic voltammetry (CV) for different situations with respect to the bulk concentrations of the ion. σCV = 0.1, T = 298 K, Ψ = I/ZFπa2cX*in(DinX (Fv/RT))1/2.

⎛ c *in + ξ c *out ⎞ n ⎛ 1 + ξje ηj−1 ⎞ 1 X ⎟ =e ⎜ X ⎟ ∏ ⎜⎜ η η ⎟ ⎝ 1 + ξ1e 1 ⎠ j = 2 ⎝ 1 + ξje j ⎠ ηn

(11)

where 4 ξj = π

application of a nth potential step in the period 0 ≤ tn ≤ τn within a series of pulses corresponding to a generic 18251

DXout DXin

πDXout t j a

(12)



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Figure 3. Comparison of the analytical (eq 13) and numerical (Section 2.2) solutions in cyclic square-wave voltammetry (SWV) for different situations with respect to the bulk concentrations of the ion. σSWV = 0.1, ΔϕSWV = 25 mV, Δϕstep = 15 mV, and T = 298 K.

with tj = ∑nl=jtl. From the expressions for the interfacial concentrations, the following solution is derived for the current of the nth pulse In =

1/2 for CV (where v is the scan rate) and σSWV = a((2f)/(Dout X )) for SWV (where f is the frequency). Thus, hereon the electrochemical response in CV and SWV will be analyzed on the basis of these two influences. In Figure 2, the results obtained with the analytical solution (eq 13) and numerical methods (Section 2.2) in CV are compared under different experimental situations with respect to the initial presence of the target ion in the inner and outer solutions. As can be observed, in all cases the shape and position of the CVs are well-described by the analytical theory presented here for σCV ≤ 0.1. In all cases the divergence found between analytical and numerical results in the position of the voltammograms is εη = |ηnum − ηanal| ≈ 0.32 (ca. 8 mV at T = 298 K), which is sufficiently close agreement for the usual experimental requirements. Thus, when the ion is initially present in the outer solution (Figure 2A,B), the forward scan toward negative potentials leads to a sigmoidal curve associated with the ion ingress under strongly convergent diffusion due to the small size of the capillary. The linear-like diffusion that dominates inside the capillary leads to a peak in the reverse scan related to the ion egress. These striking features in the CVs have been previously reported experimentally at microinterfaces.22−24 It is also worth noting that the magnitude of the reverse peak does not differ significantly between the cases c*X in = 0 (Figure 2A) and c*X out = cX*in (Figure 2B). Under these conditions the ion egress is dominated by the amount of ion ingressed in the forward scan rather than that initially present in the inner solution. This can also be deduced from the analytical solution (eq 13) because for micro- and nanocapillaries the ξ1 value is expected to be very large and then ξ1c*X out ≫ c*X in such that the influence of c*X in tends to be negligible unless the ion in the inner solution is present at a much larger concentration. Nevertheless, both

⎧ ⎛ *in + ξ c *out ⎞ n ⎪ 1 X ⎟∏ + 1 e η1 ⎠ j = 2 ξ ⎝ 1 ⎩

c ΖF 4DXout a⎨e ηn⎜ X ⎪

⎫ ⎛ 1 + ξje ηj−1 ⎞ ⎪ out * ⎟ − × ⎜⎜ c X ⎬ ηj ⎟ ⎪ ⎝ 1 + ξje ⎠ ⎭

(13)

The previous solution enables us to simulate the response associated with a reversible, simple ion transfer in any multipulse voltammetric technique, including SWV and cyclic staircase voltammetry. Note that the latter adequately describes the response in CV when the staircase potential is small enough.19 2.2. Numerical Simulations. The previous problem has also been solved by numerical methods to assess the validity of the analytical theoretical treatment. The alternating direction implicit (ADI) finite difference method has been employed with equal time steps, unequal spatial grid in the (r,z) domain with high density of nodes at the electrode surface and edge, and three-point difference approximations for the spatial derivatives.20,21

3. RESULTS AND DISCUSSION 3.1. CV and SWV Signals. Assessment of the Analytical Solutions. The analytical solution shows that the behavior of the system is defined by the dimensionless parameter (eq 12) and so by the ratio between the diffusion coefficients (DinX /Dout X ) and the dimensionless radius of the 1/2 capillary, which will be defined as σCV = a((vF)/(Dout X RT)) 18252



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situations (c*X out ≠ 0, c*X in = 0 and c*X out ≈ c*X in) can be discriminated from the current at the beginning of the scan, which is not null for c*X in ≠ 0 (Figure 2B). When the ion is initially present only in the inner solution (Figure 2C), a single peak is obtained when scanning toward positive potentials, which corresponds to the ion egress under linear-like diffusion. Indeed, it is found that the peak current scales with the square root of the scan rate. No signal is observed in the reverse scan due to the very fast diffusion of the ion toward the bulk outer solution that leads to the depletion of the species around the interface such that the ion ingress signal becomes null. In cyclic SWV (Figure 3), the typical peak-shaped response is obtained in both the forward and reverse scans for any bulk concentrations of the ion. In contrast with the behavior at macrointerfaces, the forward and backward peaks are not symmetrical with respect to the potential axis, and the peak height of the reverse scan is dependent on the vertex potential. (See Section 3.3.) As discussed for CV, the position of the voltammograms is well-described by the analytical solution (eq 13) when σSWV ≤ 0.1, with a difference between analytical and numerical results for the peak potentials smaller than εη < 0.32 (ca. 8 mV at T = 298 K) for typical SWV conditions: ΔϕSWV = 10−50 mV and Δϕstep = 2−10 mV at T = 298 K. The SWV curves cannot be unambiguously related to the ion ingress or egress due to the subtractive nature of the signal. To do this, the analysis of the forward and backward components (Figure 3D−F), is necessary and it reveals that when the ion is present in the outer solution (Figure 3A−E) the forward peak in the scan toward negative potentials is associated with the ion ingress and the peak obtained in the reverse scan with the ion egress. When the ion is present only in the inner solution (Figure 3C,F), a peak appears in the reverse scan (unlike in CV) and both the forward and backward peaks are related to the ion egress. 3.2. Influence of the System Variables. In ion-transfer processes, the diffusivity of the target species in both phases can differ significantly due to the very different properties of the media. This is, for example, the case of ion transfers between conventional solvents and liquid membranes.25 The analytical expression obtained (eq 13) enables us to investigate the influence of the ion diffusion coefficients. As shown in Figures 2 and 3, the voltammetric signal shifts to more negative potentials as the ratio DinX /Dout X decreases, that is, because the diffusion of the ion in the inner phase is slower and so its accumulation at the inner side of the interface is higher. The opposite behavior is observed when DinX /Dout X increases. As can be inferred from the analytical treatment developed in Section 2, the capillary diameter is the second main variable that defines the response, and its effect is linked to that of the time-window of the experiment through the dimensionless parameters σCV and σSWV. Note that small σ values relate to small capillary radii or long experiments (i.e., slow scan rates or low frequencies) such that the decrease in σ is associated with a more efficient radial diffusion outside the capillary. This leads to a much higher interface concentration of the ion in the inner solution with respect to the outer one such that further ingress is unfavored and the voltammograms develop at more negative potentials (Figure 4). This behavior contrasts with that observed at macrointerfaces, where the position of the signal of reversible charge-transfer processes is not affected by the time scale of the experiment.12

Figure 4. Influence of the σ parameter in CV (A) and SWV (B). DinX = in Dout X , c* X = 0, ΔϕSWV = 25 mV, Δϕstep = 2 mV, T = 298 K.

3.3. Influence of the Technique Variables. As a consequence of the asymmetry of the diffusion fields inside and outside the capillary, unusual behavior is observed with respect to the influence of the technique parameters. Figure 5 shows the effect of the vertex potential (Δϕvertex) in CV and SWV at micro/nanointerfaces (left-hand graphs) and macrointerfaces (right-hand graphs). It can be observed that the Δϕvertex influence is much less apparent and even negligible at macrointerfaces, whereas it significantly affects the magnitude of the reverse peak at micro/nanocapillaries: the more negative the Δϕvertex value, the greater the reverse peak current. The influences of the pulse amplitude (ΔϕSWV) and the potential step (Δϕstep) in SWV are studied in Figure 6. Similarly to the case of macrointerfaces, the increase in ΔϕSWV gives rise to higher and broader peaks. Thus, for very large ΔϕSWV values (>100 mV), a plateau is obtained rather than a peak, whereas for smaller pulse amplitudes (≤50 mV) the forward and backward peaks are well-defined. With regard to the influence of the potential step (Figure 6C,D), this is not significant for reversible transfers at macrointerfaces (Figure 6D), but this is not the case at micro/nanointerfaces. As can be observed in Figure 6C, the peak potentials shift toward less 18253



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Figure 5. Influence of the vertex potential in CV and SWV at micro/nanointerfaces (σCV = σSWV = 0.03; A,C) and at macrointerfaces (B,D). DinX = in in 2 in 1/2 Dout X , c* X = 0, ΔϕSWV = 25 mV, Δϕstep = 2 mV, T = 298 K. Ψ = I/ZFπa c* X (DX (Fv/RT)) .

negative potentials as the step potential is increased. For the conditions considered in the Figure, a ca. 15 mV shift is observed between Δϕstep = 2 and 15 mV. 3.4. Determination of the Ion-Transfer Potential. According to the results discussed in previous sections, the use of CV and SWV in combination with the analytical solutions presented here for the determination of the transfer formal potential (Δinoutϕ0′) should be based on the position of the voltammograms. This will be monitored through the forward peak potential in SWV (Δϕf,SWV peak ) and through the halfpeak potential of the forward scan (Δϕf,CV 1/2 ) and the peak potential in the reverse scan (Δϕr,CV peak ) in CV. When the ion is initially present in the outer solution (c*X out ≠ 0), the analytical results lead to the following expression for the half-wave potential of the forward scan in CV f,CV in 0 ′ Δϕ1/2 = Δout ϕ −

RT RT ln ξCV − 0.47 ΖF ΖF

Also, as could be observed in previous Figures, the shape of the CVs does not change significantly with the different variables studied such that the peak potential in the reverse scan can be related to Δϕf,CV 1/2 according to r,CV f,CV Δϕpeak = Δϕ1/2 + 1.44

(16)

With regard to SWV, the peak potential also varies linearly with ln ξSWV (eq 17), where ξSWV includes the influence of the intrinsic variables of the system (eq 18). However, it is difficult to find a general expression for the exact value of the peak potential given that this also depends on the step potential and the pulse amplitude f,SWV in 0 ′ Δϕpeak = Δout ϕ −

RT RT ln ξSWV − f (Δϕstep , ΔϕSWV ) ZF ZF (17)

(14)

with

where: ξCV =

RT ΖF

4 πa

DXout DXin

DXout RT vF

ξSWV =

(15) 18254

4 πa

DXout DXin

DXout 2f

(18)



Article

Figure 6. Influence of the pulse amplitude (A,B) and step potential (C,D) in SWV at micro/nanointerfaces (σSWV = 0.03; A,C) and at *in = 0, ΔϕSWV = 25 mV (C,D), Δϕstep = 5 mV (A,B). T = 298 K, Ψ = I/ZFπa2 cX*in(DinX (Fv/RT))1/2. macrointerfaces (B,D). DinX = Dout X , cX

For typical SWV conditions (Δϕstep = 2 mV, ΔϕSWV = 25 mV, T = 298 K), the value f(Δϕstep, ΔϕSWV) = 1.44 is obtained from the analytical solution. The previuos expressions enable facile prediction of the shift of the voltammograms with the capillary size, experiment time scale, and diffusivities of the ion. Also, provided that the capillary diameter and diffusion coefficients of the ion are known (for example, by means of chronoamperometric experiments18), the f,SWV Δϕf,CV 1/2 and Δϕpeak values allow for the determination of the iontransfer formal potential, and the consistency of the value obtained can be easily tested by varying the scan rate or the frequency.

simple analytical solution within 8 mV error as well as the effect on the voltammograms of the ion diffusion coefficients, the capillary size, and the time scale of the experiment. In general, any factor that promotes the accumulation of the ion at the inner side of the interfaces (i.e., slow ion diffusivity in the inner solution, small capillary radius, or long experiments) gives rise to the shift of the signal to higher overpotential with respect to the ion ingress. Thus, the position of the voltammograms varies with the scan rate in CV and the frequency in SWV, in contrast with the behavior of reversible transfers at macrointerfaces.

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AUTHOR INFORMATION

Corresponding Authors

4. CONCLUSIONS By means of a simple analytical treatment, an approximate closed-form expression has been obtained for the electrochemical response of a reversible ion transfer at micro/ nanocapillaries in any multipulse voltammetric technique. The analytical results have been assessed by comparison with numerical data under CV and SWV conditions, and simple expressions have been obtained for the determination of the ion-transfer potential. In a wide range of situations, it has been found that the shape and position of the voltammograms are well-described by the

*A.M.: Tel: +34 868 88 7524. Fax: +34 868 88 4148. E-mail: [email protected]. *R.G.C.: Tel: +44 (0) 1865 275 413. Fax: +44 (0) 1865 275 410. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.M. and E.L. greatly appreciate the financial support provided by the Ministerio de Economia y Competitividad (Project 18255



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(19) Molina, A.; Serna, C.; Camacho, L. Conditions of Applicability of the Superposition Principle in Potential Multipulse Techniques: Implications in the Study of Microelectrodes. J. Electroanal. Chem. 1995, 394, 1−6. (20) Compton, R. G.; Laborda, E.; Ward, K. R. Understanding Voltammetry: Simulation of Electrode Processes; Imperial College Press: London, 2014. (21) Britz, D. Digital Simulation in Electrochemistry, 3rd ed.; Springer: Berlin, 2005. (22) Stewart, A. A.; Taylor, G.; Girault, H. H.; McAleer, J. Voltammetry at MicroITIES Supported at the Tip of a Micropipette: Part I. Linear Sweep Voltammetry. J. Electroanal. Chem. 1990, 296, 491−515. (23) Herzog, G.; Beni, V. Stripping Voltammetry at Micro-Interface Arrays: A Review. Anal. Chim. Acta 2013, 769, 10−21. (24) Strutwolf, J.; Scanlon, M. D.; Arrigan, D. W. M. The Performance of Differential Pulse Stripping Voltammetry at MicroLiquid−Liquid Interface Arrays. J. Electroanal. Chem. 2010, 641, 7−13. (25) Molina, A.; Torralba, E.; Serna, C.; Ortuño, J. Analytical Solution for the Facilitated Ion Transfer at the Interface between Two Immiscible Electrolyte Solutions via Successive Complexation Reactions in any Voltammetric Technique: Application to Square Wave Voltammetry and Cyclic Voltammetry. Electrochim. Acta 2013, 106, 244−257.

Number CTQ2012-36700, cofunded by European Regional Development Fund). E.L. also acknowledges the funding received from the European Union Seventh Framework Programme-Marie Curie COFUND (FP7/2007-2013) under UMU Incoming Mobility Programme ACTion (U-IMPACT) Grant Agreement 267143.

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REFERENCES

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