Facing non-bilinearity in the multivariate analysis of voltammetric data

1. Facing non-bilinearity in the multivariate analysis of voltammetric data José Manuel Díaz-Cruz *, Cristina Ariño, Miquel Esteban Electroanalysis Group Department of Analytical Chemistry. University of Barcelona

2. Power source RE I AE AE WE WE WE AE RE RE V Voltammetric measurements: current vs. potential I / A E / V

3. + ne- Ox. Red. Current proportional to the flux of species + - + - + - + + - + - + - • Double layer: 1 – 10 nm • electron transfer • adsorption / desorption • Bulk solution: • (unaffected by the electrochemical process) • homogeneous reactions Intricate relationship between current and bulk concentration of species • Diffusion layer: 1 mm – 1mm • (affected by the electrochemical process) • mass transport • homogeneous reactions • Electrode • difusion / accumulation - - - + -

4. Ex: Differential Pulse Polarography, DPP Applied potencial Signal: voltammogram tp I = I2 – I1 = I (E+DE) – I (E) E 2 W1/2 Ep Ip a c* tp≈ 50 ms td ≈ 1 s Ep ≈ 50 mV 1 ½Ip t td E Ep on: Usually, signals have characteristic shapes and a rigorous theoretical model available (Hard modelling)

5. chemical species I / A component single electrochemical process single electrochemical signal (usually, a peak) shape constraints E / V peaks stay at the same potential In many cases voltammetric data are bilinear: However, in the presence of many overlapping signals, multivariate analysis is required (Soft modelling) (Cd2+ + Pb2+ + PC5 system) ... thus bilinear methods like MCR-ALS can be applied

6. MCR-ALS scheme

7. I / A pH I / A E / V sv E / V component ... but sometimes voltammetric data are non-bilinear : This can be noticed by: - movement of signals along E axis - changes in signal width or symmetry - too high number of components by SVD - too high lof by MCR-ALS (Cd2+ + PC2 system 1:4) And can be due to: - fast equilibrium between electroactive species - changes in electrochemical reversibility - changes in homogeneous reaction kinetics

8. I DE (one for every voltammogram) E Approaches to deal with non-linearity: • Some ideas: • In matrices with signals moving along the potential axis, the corresponding pure signals • have to move also to keep I = C VT at every row of the I matrix. • The movements have to leave the position of the other signals unchanged. • The movements are measured from an arbitrary reference position in the form of DE • (potential shifts) values, one for every voltammogram. puresignals

9. pure signals without DE would yield a (bilinear) corrected matrix: I I E E Approaches to deal with non-linearity: • Some ideas: • In matrices with signals moving along the potential axis, the corresponding pure signals • have to move also to keep I = C VT at every row of the I matrix. • The movements have to leave the position of the other signals unchanged. • The movements are measured from an arbitrary reference position in the form of DE • (potential shifts) values, one for every voltammogram. Thus, we have to find DE !

10. integration with the rest of signals using a common E axis (only red signal) extrapolated points interpolated points (splines) from E to E’ shift of E axis (only red signal) DE E’ E’ Programs to deal with non-linearity: shiftfit program (for signals moving along potential axis during the experiment) - Uses pure voltammograms of any shape which are kept constant along the matrix except for the height and position, which are least-squares optimised, row by row. I I shiftfit E E

11. The algorithm is somewhat more involved… (shiftfit/shiftcalc programs based upon the Matlab command lsqcurvefit) Analyst 133 (2008) 112

12. experimental voltammogram estimated DE for every component optimised DE for every component contains all parameters not to be optimised lower and upper possible values of DE 1 or 0 to indicate if the component moves estimated concentrations reference pure signals invokes another program line where max. number of iterations and tolerances are specified [Irep]=shiftcalc(delta,cv) options=optimset('Display','off', 'Diagnostics','off', 'LevenbergMarquardt','on', 'MaxIter',50,'TolX',0.001, 'TolFun',0.001) invokes external shiftcalc function to iteratively try delta values until the resulting matrix (reproduced) approaches the experimental one How does lsqcurvefit works? Inside shiftfit, for every voltammogram: [delta]=lsqcurvefit('shiftcalc',delta0,cv,Iexp,lv,uv,options);

13. shiftcalc shiftfit Analyst 133 (2008) 112

14. Example of shiftfit application: Zn2+- glycine system Analyst 133 (2008) 112 lof. 15.7 % lof. 6.7 % b1, b2values closer to Literature than those obtained by MCR-ALS

15. pHfit program (for the especially involved evolution of signals with pH) - Uses pure voltammograms of any shape which are kept constant along the matrix except for the height and position, which are least-squares optimised, but not row by row. Instead, DE values are given by a parametric equation as a function of pH whose parameters are least-squared optimised. - Parametric equations can be: straight line sigmoid (signals can be also immobile or randomly moving with pH) Analyst 135 (2010) 1653

16. The algorithm: Analyst 135 (2010) 1653

17. PCn γ-Glu Cys Gly experimental matrix svd exp. Example of pHfit application: Cd2+- PC2 1:4 at different pH values: cor. reproduced matrix concen- trations DE vs. pH reference signals lof. 16.7 % corrected matrix Analyst 135 (2010) 1653

18. I c a E related to b related to d GPA program (for moving signals which also change width and symmetry) (Gaussian Peak Adjustment) - Uses peak-shaped pure voltammograms following a double-gaussian parametric function whose parameters are least-squares adjusted row by row and provide the height, area, width and symmetry of the signals. - The parametric equations is: left side of maximum: right side of maximum: - Along the rows, the optimised values of a, b, c, d are used as estimations for the next row calculations Anal. Chim. Acta 689 (2011) 198

19. The algorithm: Anal.Chim.Acta 689 (2011) 198

20. Example of GPA application: PC5 at different pH values lof. 5.4 % Anal.Chim.Acta 689 (2011) 198 experimental matrix svd concentrations vs. pH from currents from areas reproduced matrix error matrix w1/2vs. pH (width) DE vs. pH

21. Comparing different approaches: Anal.Chim.Acta 689 (2011) 198 Zn2+- oxalate system at increasing oxalate concentrations shiftfit 1 comp. (reproduced and error matrices) experimental matrix svd lof. 17.8 % lof. 6.6 % lof. 4.6 % MCR-ALS 2 comp. (reproduced and error matrices) GPA 1 comp. (reproduced and error matrices)

23. Application of other parametric functions in programs analogue to GPA • Implementation of constraints along the different voltammograms in • GPA program • (e.g. sigmoid/linear evolution of potentials, chemical equilibrium …) • Fitting of parametric functions involving both variables in data matrices • (mostly in data consisting of currents vs. potential and time) Analyst 136 (2011) 4696 Present and future trends: (asymmetric logistic function) Free download and additional information about the programs at: http://www.ub.edu/dqaelc/programes_eng.html