Square-Wave Voltammetry

Abstract

Square-wave voltammetry (SWV) is one of the four major voltammetric techniques provided by modern computer-controlled electroanalytical instruments, such as Autolab and μAutolab (both EcoChemie, Utrecht), BAS 100 A (Bioanalytical Systems), and PAR Model 384 B (Princeton Applied Research) [1]. The other three important techniques are single scan and cyclic staircase, pulse, and differential pulse voltammetry (see Chap. II.2). All four are either directly applied or after a preconcentration to record the stripping process. The application of SWV boomed in the last decade, first because of the widespread use of the instruments mentioned above, second because of a well-developed theory, and finally, and most importantly, because of its high sensitivity to surface-confined electrode reactions. Adsorptive stripping SWV is the best electroanalytical method for the determination of electroactive organic molecules that are adsorbed on the electrode surface [2].

Appendix

(A) For a stationary planar diffusion model of a simple redox reaction (Eq. II.3.1) the following differential equations and boundary conditions can be formulated:

$\frac{{\partial c_{{\rm{red}}} }}{{\partial t}} = D_{\rm{r}} \frac{{\partial ^2 c_{{\rm{red}}} }}{{\partial x^2 }}$

((II.3.15))

$\frac{{\partial c_{{\rm{ox}}} }}{{\partial t}} = D_{\rm{o}} \frac{{\partial ^2 c_{{\rm{ox}}} }}{{\partial x^2 }}$

((II.3.16))

$t = 0,\, x \geq 0:\ c_{\rm red} = c^*,\, c_{\rm ox} = 0$

((II.3.17))

$t > 0,\, x \rightarrow \infty,\ c_{\rm red} \rightarrow c^*,\, c_{\rm ox} \rightarrow 0$

((II.3.18))

$x = 0:\ D_{\rm r} \left( {\frac{{\partial c_{{\rm{red}}} }}{{\partial x}}} \right)_{x = 0} = \frac{i}{{nFS}}$

((II.3.19))

$D_{\rm o} \left( {\frac{{\partial c_{{\rm{ox}}} }}{{\partial x}}} \right)_{x = 0} = - \frac{i}{{nFS}}$

((II.3.20))

If reaction (II.3.1) is fast and reversible, the Nernst equation has to be satisfied:

$(c_{\rm ox})_{x=0} = (c_{\rm red})_{x = 0} \exp (\varphi)$

((II.3.21))

$\varphi = \frac{nF}{RT} (E - E^{{\,{\scriptscriptstyle\bigcirc\raisebox{1.2pt}{$\rule{7.5pt}{0.4pt}$}}}})$

((II.3.22))

If reaction (II.3.1) is kinetically controlled, the Butler-Volmer equation applies:

$\frac{i}{nFS} = -k_{\rm s} \exp (- \alpha \varphi) [(c_{\rm ox})_{x = 0} -(c_{\rm{red}})_{x = 0}\exp (\varphi)]$

((II.3.23))

where c _red and c _ox are the concentrations of the reduced and oxidized species, respectively. D _r and D _o are the corresponding diffusion coefficients, k _s is the standard rate constant, α is the transfer coefficient, $E^{{\,{\scriptscriptstyle\bigcirc\raisebox{1.2pt}{$\rule{7.5pt}{0.4pt}$}}}}$ is the standard potential, x is the distance from the electrode surface, t is the time variable, and the other symbols are explained below Eq. (II.3.1) above.

The solution of Eqs. (II.3.15), (II.3.16), (II.3.17), (II.3.18), (II.3.19), (II.3.20), (II.3.21) and (II.3.22) is an integral equation [120]:

$\int\limits_0^t \varPhi ^* [\pi (t - \tau)]^{-1/2} {\textrm{d}} \tau = \exp (\varphi ^*) [1 + \exp (\varphi ^*)]^{-1}$

((II.3.24))

where

$\varPhi ^* = i [nFSc^*D_{\rm r}^{1/2}]^{-1}$

((II.3.25))

$\varphi ^* = nF(E - E_{1/2})/RT$

((II.3.26))

$E_{1/2} = E^{\,{\scriptscriptstyle\bigcirc\raisebox{1.2pt}{$\rule{7.5pt}{0.4pt}$}}} + RT[\ln (D_{\rm r}/D_{\rm o})]/2nF$

((II.3.27))

The solution for the kinetically controlled reaction is [120]:

$\begin{array}{l}\displaystyle \varPhi ^* = - \uplambda ^*\exp \left( { - \alpha \varphi ^*} \right)\left[ {1 + \exp \left( {\varphi ^*} \right)} \right] \\ \noalign{}\qquad\,\,\,\, \int\limits_0^t \displaystyle \varPhi ^*\left[ {\pi \left( {t - \tau } \right)} \right]^{ - 1/2} {\textrm{d}}\tau + \uplambda ^*\exp \left[ {\left( {1 - \alpha } \right)\varphi ^*} \right]\end{array}$

((II.3.28))

The convolution integrals in Eqs. (II.3.24) and (II.3.28) can be solved by the method of numerical integration proposed by Nicholson and Olmstead [38, 121]:

$\int\limits_0^t {\varPhi ^*\left[ {\pi \left( {t - \tau } \right)} \right]^{ - 1/2} } {\textrm{d}}\tau = 2\left( {d/\pi } \right)^{1/2} \sum\limits_{j = 1}^m {\varPhi _j^* } S_{m - j + 1}$

((II.3.29))

where d is the time increment, $t = md,\ \varPhi_j^*$ is the average value of the function Φ ^* within the j ^th time increment, $S_{\rm k} = k^{1/2} - (k - 1)^{1/2}$ and S ₁ = 1. Each square-wave half-period is divided into 25 time increments: d = (50 f)⁻¹. By this method, Eq. (II.3.24) is transformed into the system of recursive formulae:

$\varPhi _m = 5\left( {\pi /2} \right)^{1/2} \exp \left( {\varphi _m^* } \right)\left[ {1 + \exp \left( {\varphi _m^* } \right)} \right]^{ - 1} -\sum\limits_{j = 1}^{m - 1} {\varPhi _j } S_{m - j + 1}$

((II.3.30))

where $\varPhi = i [nFSc^*(D_{\rm r}f)^{1/2}]^{-1},\, \varphi_m^* = nF(E_m - E_{1/2})/RT,\, m = 1, 2, 3, \ldots {\rm M}$ and ${\rm M} = 50\, (E_{\rm fin} - E_{\rm st})/\Delta E$. The potential E _m changes from E _stair = E _st to E _stair = E _fin according to Fig. II.3.1. The recursive formulae for the kinetically controlled reaction are [38, 40, 41, 44]:

$\varPhi_m = Z_1 - Z_2 \sum\limits_{j = 1}^{m-1} \varPhi_j S_{m-j+1}$

((II.3.31))

$Z_1 = \frac{{\uplambda \exp [(1 - \alpha )\varphi _m^{\rm{*}} ]}}{{1 + \frac{{\uplambda \sqrt 2 }}{{5\sqrt {\rm{\pi }} }}[\exp ( - \alpha \varphi _m^{\rm{*}} ) + \exp ((1 - \alpha )\varphi _m^{\rm{*}} )]}}$

((II.3.32))

$Z_2 = \frac{{\frac{{\uplambda \sqrt 2 }}{{5\sqrt {\rm{\pi }} }}[\exp ( - \alpha \varphi _m^{\rm{*}} ) + \exp ((1 - \alpha )\varphi _m^{\rm{*}} )]}}{{1 + \frac{{\uplambda \sqrt 2 }}{{5\sqrt {\rm{\pi }} }}[\exp ( - \alpha \varphi _m^{\rm{*}} ) + \exp ((1 - \alpha )\varphi _m^{\rm{*}} )]}}$

((II.3.33))

where

$\lambda = \frac{{k_{\textrm{s}}}}{{\sqrt {D_{\textrm{o}} {\textrm{f}}}}}\left({\frac{{D_{\textrm{o}}}}{{D_{\textrm{r}}}}}\right)^{\frac{\alpha }{2}}$

is a dimensionless kinetic parameter.

(B) On a stationary spherical electrode, a simple redox reaction

${\rm Ox} + n{\textrm{e}}^- \leftrightarrows {\rm Red}$

((II.3.34))

can be mathematically represented by the well-known integral equation [120]:

$\varPhi = \frac{{k_{\rm{s}} }}{{(Df)^{1/2} }}\exp ( - \alpha \varphi )[1 - f^{1/2} (1 + \exp (\varphi ))I^{\rm{o}} ]$

((II.3.35))

$\varPhi = i(nFSc_{\rm ox}^*)^{-1}(Df)^{-1/2}$

((II.3.36))

$I^{\rm o} = \int\limits_0^t \varPhi [\pi (t-u)]^{-1/2} {\textrm{d}}u - \frac{D^{1/2}}{r} \int\limits_0^t \varPhi \exp [D(t -u)r^{-2}] {\textrm{erfc}} [D^{1/2}r^{-1} (t - u)^{1/2}]{\rm{d}}u$

((II.3.37))

where r is the radius of the spherical electrode and $c_{{\rm{ox}}}^{\rm{*}}$ is the bulk concentration of the oxidized species. The meanings of all other symbols are as above. It is assumed that both the reactant and product are soluble, that only the oxidized species is initially present in the solution, and that the diffusion coefficients of the reactant and product are equal. For numerical integration, Eq. (II.3.35) can be transformed into a system of recursive formulae [51]:

$\varPhi_{\rm m} = \frac{{ - \frac{{D^{1/2} }}{{r^{} f^{1/2} }} - \left( {1 + \exp (\varphi _m )} \right)\sum\limits_{i = 1}^{m - 1} {\varPhi _i S_{m - i + 1} } }}{{\frac{D}{{k_{\rm{s}} r}}\exp (\alpha \varphi _m ) + S_1 \left( {1 + \exp (\varphi _m )} \right)}}$

((II.3.38))

$S_1 = 1 - \exp (Df^{-1}r^{-2}N^{-1}) {\textrm{erfc}}(D^{1/2}f^{-1/2}r^{-1}N^{-1/2})$

((II.3.39))

$\begin{array}{l}\displaystyle S_k = \exp \left[ {Df^{ - 1} r^{ - 2} N^{ - 1} \left( {k - 1} \right)} \right]{\textrm{erfc}}\left[ {D^{1/2} f^{ - 1/2} r^{ - 1} N^{ - 1/2} \left( {k - 1} \right)^{1/2} } \right] \\ \noalign{}\displaystyle \qquad\, - \exp \left( {Df^{ - 1} r^{ - 2} N^{ - 1} k} \right){\textrm{erfc}}\left( {D^{1/2} f^{ - 1/2} r^{ - 1} N^{ - 1/2} k^{1/2}} \right)\end{array}$

((II.3.40))

where N is the number of time increments in each square-wave period. The ratio k _s r/D is the dimensionless standard charge transfer rate constant of reaction (II.3.34) and the ratio rf ^1/2/D ^1/2 is the dimensionless electrode radius.

(C) A surface redox reaction (II.3.4) on a stationary planar electrode is represented by the system of differential equations (II.3.15) and (II.3.16), with the following initial and boundary conditions [89]:

$t= 0,\, x \geq 0\,:\, c_{\rm ox} = c_{\rm ox}^*,\, c_{\rm red} = 0,\ \varGamma_{\rm ox} = \varGamma_{\rm red} = 0$

((II.3.41))

$t > 0\,:\, x \rightarrow \infty:\ c_{\rm ox} \rightarrow c_{\rm ox}^*,\ c_{\rm red} \rightarrow 0$

((II.3.42))

$x = 0\,:\, K_{\rm ox}(c_{\rm ox})_{x=0} = \varGamma _{\rm ox}$

((II.3.43))

$K_{\rm red} (c_{\rm red})_{x=0} = \varGamma_{\rm red}$

((II.3.44))

$i/nFS = k_{\rm s} \exp (-\alpha\varphi)[\varGamma_{\rm ox} - \exp (\varphi) \varGamma_{\rm red}]$

((II.3.45))

$D_{\rm o} (\partial c_{\rm ox}/\partial x)_{x=0} = d \varGamma_{\rm ox}/dt + i/nFS$

((II.3.46))

$D_{\rm r}(\partial c_{\rm red}/\partial x)_{x = 0} = d\varGamma_{\rm red}/dt-i/nFS$

((II.3.47))

$\varphi = nF(E - E^{\,{\scriptscriptstyle\bigcirc\raisebox{1.2pt}{$\rule{7.5pt}{0.4pt}$}}}_{\varGamma_{\rm ox}/\varGamma_{\rm red}})/RT$

((II.3.48))

$E_{{\varGamma_{\rm ox}}/{\varGamma_{\rm red}}}^{{\,{\scriptscriptstyle\bigcirc\raisebox{1.2pt}{$\rule{7.5pt}{0.4pt}$}}}} = E^{{\,{\scriptscriptstyle\bigcirc\raisebox{1.2pt}{$\rule{7.5pt}{0.4pt}$}}}} + (RT/nF) \ln (K_{\rm red}/K_{\rm ox})$

((II.3.49))

where Γ _ox and Γ _red are the surface concentrations of the oxidized and reduced species, respectively, and K _ox and K _red are the constants of linear adsorption isotherms. The solution of Eqs. (II.3.15) and (II.3.16) is an integral equation:

$\begin{array}{l}\displaystyle i/nFS = k_{\rm{s}} \exp \left( { - \alpha \varphi } \right)\\ \noalign{} \displaystyle \left\{ {K_{{\rm{ox}}} c_{{\rm{ox}}}^* \left[ {1 - \exp \left( {D_{\rm{o}} tK_{{\rm{ox}}}^{ - 2} } \right){\textrm{erfc}}\left( {D_{\rm{o}}^{1/2} t^{1/2} K_{{\rm{ox}}}^{ - 1} } \right)} \right] - I_{{\rm{ox}}} - I_{{\rm{red}}} \exp \left( \varphi \right)} \right\}\end{array}$

((II.3.50))

$I_{{\rm{ox}}} = \int\limits_0^t {\left( {i/nFS} \right)\exp \left[ {D_{\rm{o}} \left( {t - \tau } \right)K_{{\rm{ox}}}^{ - 2} } \right]} {\textrm{erfc}}\left[ {D_{\rm{o}}^{1/2} \left( {t - \tau } \right)^{1/2} K_{{\rm{ox}}}^{ - 1} } \right]{\textrm{d}}\tau$

((II.3.51))

$I_{{\rm{red}}} = \int\limits_0^t {\left( {i/nFS} \right)\exp \left[ {D_{\rm{r}} \left( {t - \tau } \right)K_{{\rm{red}}}^{ - 2} } \right]} {\textrm{erfc}}\left[ {D_{\rm{r}}^{1/2} \left( {t - \tau } \right)^{1/2} K_{\rm{red}}^{ - 1} } \right]{\textrm{d}}\tau$

((II.3.52))

For numerical integration, Eqs. (II.3.50), (II.3.51) and (II.3.52) are transformed into a system of recursive formulae [93]:

$\varPhi_{\rm m} = \frac{{\kappa \exp ( - \alpha \varphi _m )\left[ {1 - \exp \left( {a_{{\rm{ox}}}^{ - 2} mN^{ - 1} } \right){\textrm{erfc}}\left( {a_{{\rm{ox}}}^{ - 1} m^{1/2} N^{ - 1/2} } \right) - SS_1 + SS_2 } \right]}}{{1 + \kappa \exp ( - \alpha \varphi _m )\left[ {2(N\pi )^{ - 1/2} \left( {a_{{\rm{ox}}} + a_{{\rm{red}}} \exp (\varphi _m )} \right) - a_{{\rm{ox}}}^2 M_1 - a_{{\rm{red}}}^2 \exp (\varphi _m )P_1 } \right]}}$

((II.3.53))

$SS_1 = 2\left( {N\pi } \right)^{-1/2} \left[ {a_{{\rm{ox}}} + a_{{\rm{red}}} \exp \left( {\varphi _m } \right)} \right]\sum\limits_{j = 1}^{m - 1} {\varPhi _j S_{m - j + 1} }$

((II.3.54))

$SS_2 = \sum\limits_{j = 1}^{m - 1} {\varPhi _j } \left[ {a_{{\rm{ox}}}^2 M_{m - j + 1} + a_{{\rm{red}}}^2 \exp \left( {\varphi _m } \right)P_{m - j + 1} } \right]$

((II.3.55))

$\varPhi = i\left( {nFSK_{{\rm{ox}}} c_{{\rm{ox}}}^* f} \right)^{ - 1}$

((II.3.56))

$\kappa = k_{\textrm{s}} /f$

((II.3.57))

$a_{{\rm{ox}}} = K_{{\rm{ox}}} f^{1/2} D_{\rm{o}}^{{\rm{ - 1/2}}} $

((II.3.58))

$a_{{\rm{red}}} = K_{{\rm{red}}} f^{1/2} D_{\rm{r}}^{{\rm{ - 1/2}}} $

((II.3.59))

$d = N^{ - 1} f^{ - 1} $

((II.3.60))

$S_k = k^{1/2} - \left( {k - 1} \right)^{1/2}$

((II.3.61))

$M_1 = 1 - \exp \left( {a_{{\rm{ox}}}^{ - 2} N^{ - 1} } \right){\textrm{erfc}}\left( {a_{{\rm{ox}}}^{ - 1} N^{ - 1/2} } \right)$

$\begin{array}{l}\displaystyle M_k = \exp \left[ {a_{{\rm{ox}}}^{{\rm{ - 2}}} \left( {k - 1} \right)N^{ - 1} } \right]{\textrm{erfc}}\left[ {a_{{\rm{ox}}}^{ - 1} \left( {k - 1} \right)^{1/2} N^{-1/2} } \right] \\ \noalign{}\qquad\,\,\, \displaystyle - \exp \left[ {a_{{\rm{ox}}}^{{\rm{ - 2}}} kN^{ - 1} } \right]{\textrm{erfc}}\left[ {a_{{\rm{ox}}}^{{\rm{ - 1}}} k^{ 1/2} N^{ - 1/2} } \right]\end{array}$

((II.3.62))

$P_1 = 1 - \exp \left( {a_{{\rm{red}}}^{{\rm{ - 2}}} N^{ - 1} } \right){\textrm{erfc}}\left( {a_{{\rm{red}}}^{ - 1} N^{ - 1/2} } \right)$

$\begin{array}{l}\displaystyle P_k = \exp \left[ {a_{{\rm{red}}}^{{\rm{ - 2}}} \left( {k - 1} \right)N^{ - 1} } \right]{\textrm{erfc}}\left[ {a_{{\rm{red}}}^{{\rm{ - 1}}} \left( {k - 1} \right)^{1/2} N^{ - 1/2} } \right] \\ \noalign{}\qquad\,\,\, \displaystyle - \exp \left[ {a_{{\rm{red}}}^{ - 2} kN^{ - 1} } \right]{\textrm{erfc}}\left[ {a_{{\rm{red}}}^{{\rm{ - 1}}} k^{ 1/2} N^{ - 1/2} } \right]\end{array}$

((II.3.63))

(D) In a simplified approach to the surface redox reaction, the transport of Ox and Red in the solution is neglected. This assumption corresponds to a totally irreversible adsorption of both redox species [94]:

$({\rm Ox})_{\rm ads} + n{\textrm{e}}^- \rightleftarrows ({\rm Red})_{\rm ads}$

((II.3.64))

The current is determined by Eq. (II.3.45), with the initial and boundary conditions:

$t=0,\, \varGamma_{\rm ox} = \varGamma_{\rm ox}^*,\, \varGamma_{\rm red} = 0$

((II.3.65))

$t > 0\,:\, \varGamma_{\rm ox} +\varGamma_{\rm red} = \varGamma_{\rm ox}^* $

((II.3.66))

${\textrm{d}}\varGamma_{\rm ox}/{\textrm{d}}t = -i/nFS$

((II.3.67))

${\textrm{d}}\varGamma_{\rm red} / {\textrm{d}}t = i/nFS$

((II.3.68))

The solution of Eq. (II.3.45) is a system of recursive formulae:

$\phi_{\rm m} = \frac{{\kappa \exp ( - \alpha \varphi _m )\left[ {1 - N^{ - 1} (1 + \exp (\varphi _m ))\sum\limits_{j = 1}^{m - 1} {\varPhi _{{j}} } } \right]}}{{1 + \kappa \exp ( - \alpha \varphi _m )N^{ - 1} (1 + \exp (\varphi _m ))}}$

((II.3.69))

$\varPhi = \frac{i}{nFS\varGamma_{\rm ox}^*f}$

((II.3.70))

The kinetic parameter κ is defined by Eq. (II.3.57).

Under chronoamperometric conditions (E = const.), the solution of Eq. (II.3.45) is

$\varPhi = \uplambda \exp (- \alpha \varphi) \exp [-\uplambda \exp(-\alpha \varphi) (1 + \exp (\varphi))]$

((II.3.71))

$\varPhi = it (nFS\varGamma_{\rm ox}^*)^{-1}$

((II.3.72))

$\uplambda = k_{\rm s}t$

((II.3.73))

If $\varphi = 0$, Eq. (II.3.71) is reduced to

$i/nFS \varGamma_{\rm ox}^* = k_{\textrm{s}} \exp (-2k_{\rm s}t)$

((II.3.74))

The maximum chronoamperometric response is defined by the first derivative of Eq. (II.3.71):

$\partial \varPhi / \partial \uplambda = 0$

((II.3.75))

$\uplambda_{\rm max} = \exp(\alpha \varphi) [1 + \exp (\varphi)]^{-1}$

((II.3.76))

The second condition is:

$\partial \uplambda _{\max} / \partial \varphi = 0$

((II.3.77))

with the result:

$\exp (\varphi_{\max}) = \frac{\alpha}{1-\alpha}$

((II.3.78))

$\uplambda_{\max, \max} = \alpha^{\alpha} (1 - \alpha)^{1 - \alpha}$

((II.3.79))

$\varPhi_{\max} = \frac{1 - \alpha}{e}$

((II.3.80))

This derivation shows that, for any electrode potential E, there is a certain dimensionless kinetic parameter λ_max which gives the highest response (Eq. II.3.76). The maximum of λ_max (Eq. II.3.79) is a parabolic function of the transfer coefficient: $0.5 \leq \uplambda_{\max, \max} < 1$, for $0 < \alpha < 1$. If $\alpha = 0.5$, then $\uplambda_{\max, \max} = 0.5$ and $\varPhi_{\max} = (2{\rm e})^{-1}$. This is in the agreement with Eq. (II.3.74). From the condition ∂i/∂k _s = 0, it follows that k _s,max = (2t)⁻¹ and $(i/nFS \varGamma _{{\rm{ox}}}^*)_{\max} = (2{\rm e}t)^{-1}$.