Differential double pulse voltammetry (DDPV) and additive differential pulse voltammetry (ADPV) applied to the study of the ACDT mechanism

Abstract

The value of double potential pulse differential techniques for electrochemical studies of dynamic chemical speciation is investigated within the context of liquid|liquid electrochemistry. Then, not only information about the speciation thermodynamics and kinetics is accessible, but also about the species lipophilicity. This provides a comprehensive view of their behaviour in natural and biological media. With the above aim, the current-potential response of the ACDT mechanism (aqueous complexation-dissociation coupled to transfer), where two chemically linked species can transfer between a hydrophilic and a lipophilic phase, is modelled in differential double pulse voltammetry (DDPV) and additive differential pulse voltammetry (ADPV). Explicit closed-form expressions are deduced for both techniques at liquid|liquid macrointerfaces, applicable to any charge number and lipophilicity of the species. The DDPV and ADPV signals are shown to be well sensitive to the features of the chemical reaction, as well as to the species charge and lipophilic character. Also, these techniques offer higher resolution and discrimination against superimposed signals than direct-current techniques, together with high sensitivity and minimization of capacitive effects and background currents, which are key advantages for accurate quantitative analysis.

Appendix

Variables

A :

Area of the interface

c ^∗ :

Sum of the initial concentrations of transferable species in the aqueous phase

${c}_{\mathrm{i}}^{\ast }$ :

Initial equilibrium concentration of species i in the aqueous phase (i = X, XL, L)

${D}_{\mathrm{i}}^{\upalpha}$ :

Diffusion coefficient of species i in phase α(= w, o)

D ^α :

Diffusion coefficient of the free and complexed species in phase α (${D}^{\upalpha}={D}_{\mathrm{X}}^{\upalpha}={D}_{\mathrm{X}\mathrm{L}}^{\upalpha}$)

E ₁ :

First potential pulse applied in DDPV and ADPV

E _1/2 :

Half-wave potential

E _c :

Zero-crossing potential in ADPV

E _peak :

Peak potential in DDPV

E _valley :

Potential of the valley between two peaks in DDPV

F :

Faraday constant

I ₁ :

Current measured at the end of the first potential pulse

I ₂ :

Current measured at the end of the second potential pulse

I _ADPV :

Current response in ADPV

${I}_{\mathrm{lim}}^{(1)}$ :

Limiting current of the first wave obtained in NPV

I _lim :

Total limiting current of the NPV curve

k ₁ :

Forward rate constant (M⁻¹s^‐1)

k ₂ :

Backward rate constant (s^‐1)

K :

Apparent equilibrium constant under excess of ligand ($={K}_{\mathrm{c}}{c}_{\mathrm{L}}^{\ast }$)

K _c :

Chemical equilibrium constant based on concentrations (M⁻¹)

R :

Molar gas constant

T :

Absolute temperature

t :

Electrolysis time

z _i :

Ionic charge number of species i

${\Delta}_{\mathrm{O}}^{\mathrm{W}}{\phi}_{\mathrm{i}}^{0\prime }$ :

Formal transfer potential of species i

ΔE:

Pulse amplitude

ΔI_DDPV:

Current response in DDPV

τ :

Duration of the potential pulses applied in NPV

τ ₁ :

Duration of the first potential pulse applied in DDPV and ADPV

τ ₂ :

Duration of the second potential pulse applied in DDPV and ADPV

Functions

$$ \mathit{\operatorname{erfc}}(x)=1-\frac{2}{\sqrt{\pi }}{\int}_0^x{e}^{-{t}^2} dt $$

$$ {E}_{1/2}^{\mathrm{i}}={\Delta}_{\mathrm{O}}^{\mathrm{W}}{\phi}_{\mathrm{i}}^{0\hbox{'}}+\frac{RT}{z_{\mathrm{i}}F}\ln \sqrt{\frac{D^{\mathrm{W}}}{D^{\mathrm{O}}}};i=X, XL $$

$$ F\left({\chi}_{\mathrm{j}}\right)=\frac{\sqrt{\pi }}{2}{\chi}_{\mathrm{j}}{e}^{{\left(\frac{\chi_{\mathrm{j}}}{2}\right)}^2}\mathit{\operatorname{erfc}}\left(\frac{\chi_{\mathrm{j}}}{2}\right);j=1,2 $$

$$ {I}_{\mathrm{d}}\left({z}_{\mathrm{X}},{c}^{\ast}\right)={z}_{\mathrm{X}} FA{c}^{\ast}\sqrt{\frac{D^{\mathrm{W}}}{\pi t}} $$

$$ {I}_{\mathrm{d},2}\left({z}_{\mathrm{X}},{c}^{\ast}\right)={z}_{\mathrm{X}} FA{c}^{\ast}\sqrt{\frac{D^{\mathrm{W}}}{\pi {t}_2}} $$

$$ {k}_1^{\prime }={k}_1{c}_{\mathrm{L}}^{\ast } $$

$$ \alpha ={e}^{\eta_{1/2,\mathrm{X}}^{(2)}}\left({e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}-{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}}^{(1)}}\right)-{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}\left({e}^{\eta_{1/2,\mathrm{X}}^{(1)}}-{e}^{\eta_{1/2,\mathrm{X}}^{(2)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}\right) $$

$$ \beta =\left({e}^{\eta_{1/2,\mathrm{X}}^{(1)}}-{e}^{\eta_{1/2,\mathrm{X}}^{(2)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}\right)+\left({e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}-{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}}^{(1)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}}^{(2)}}\right) $$

$$ \varDelta {e}^{\eta_{1/2}^{\left(\mathrm{j}\right)}}={e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{\left(\mathrm{j}\right)}}-{e}^{\eta_{1/2,\mathrm{X}}^{\left(\mathrm{j}\right)}};j=1,2;i=X, XL $$

$$ {\varepsilon}_{\mathrm{j}}=1+{e}^{\eta_{1/2,\mathrm{X}}^{\left(\mathrm{j}\right)}}+K\left(1+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{\left(\mathrm{j}\right)}}\right) $$

$$ {\eta}_{1/2,\mathrm{i}}^{\left(\mathrm{j}\right)}=\frac{z_{\mathrm{i}}F}{RT}\left({E}_{\mathrm{j}}-{E}_{1/2}^{\left(\mathrm{i}\right)}\right) $$

$$ {\theta}_{\mathrm{j}}=K{e}^{\eta_{1/2,\mathrm{X}}^{\left(\mathrm{j}\right)}}+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{\left(\mathrm{j}\right)}}+{e}^{\eta_{1/2,\mathrm{X}}^{\left(\mathrm{j}\right)}}{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{\left(\mathrm{j}\right)}}\left(1+K\right) $$

$$ \kappa ={k}_1^{\prime }+{k}_2 $$

$$ \lambda =\frac{e^{\eta_{1/2,\mathrm{X}}^{(1)}}\left(1+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}\right)-\frac{z_{\mathrm{X}\mathrm{L}}}{z_{\mathrm{X}}}{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}\left(1+{e}^{\eta_{1/2,\mathrm{X}}^{(1)}}\right)}{1+{e}^{\eta_{1/2,\mathrm{X}}^{(1)}}+K\left(1+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}\right)} $$

$$ \mu =\left({e}^{\eta_{1/2,\mathrm{X}}^{(1)}}-{e}^{\eta_{1/2,\mathrm{X}}^{(2)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}\right)-\left({e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}}-{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}\right)\left(1+{e}^{\eta_{1/2,\mathrm{X}}^{(2)}}\right) $$

$$ \varphi ={e}^{\eta_{1/2,\mathrm{X}}^{(2)}}{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}\left(1+K\right)-{e}^{\eta_{1/2,\mathrm{X}}^{(1)}}{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(2)}}-K{e}^{\eta_{1/2,\mathrm{X}}^{(2)}}{e}^{\eta_{1/2,\mathrm{X}\mathrm{L}}^{(1)}} $$

$$ {\chi}_1=\frac{2{\varepsilon}_1\sqrt{\kappa t}}{\theta_1} $$

$$ {\chi}_1^{\prime }=\frac{2\sqrt{\kappa t}}{K} $$

$$ {\chi}_2=\frac{2{\varepsilon}_2\sqrt{\kappa {t}_2}}{\theta_2} $$