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

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

4. Applied potencial tptp t tdtd EpEp 1 2 E t p ≈ 50 ms t d ≈ 1 s  E p ≈ 50 mV E I p  c * EpEp ½ Ip½ Ip W 1/2 Signal: voltammogram  I = I 2 – I 1 = I (E+  E) – I (E) Ex: Differential Pulse Polarography, DPP on: Usually, signals have characteristic shapes and a rigorous theoretical model available (Hard modelling)

5. However, in the presence of many overlapping signals, multivariate analysis is required (Soft modelling) I / A E / V component single electrochemical process chemical species single electrochemical signal (usually, a peak) shape constraints... thus bilinear methods like MCR-ALS can be applied In many cases voltammetric data are bilinear : peaks stay at the same potential (Cd 2+ + Pb 2+ + PC 5 system)

6. MCR-ALS scheme

7. I / A E / V pH... 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 And can be due to: - fast equilibrium between electroactive species - changes in electrochemical reversibility - changes in homogeneous reaction kinetics component sv (Cd 2+ + PC 2 system 1:4) I / A E / V

8. I 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 V T 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  E (potential shifts) values, one for every voltammogram.  E (one for every voltammogram) pure signals

9. I 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 V T 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  E (potential shifts) values, one for every voltammogram. pure signals without  E would yield a (bilinear) corrected matrix: I E Thus, we have to find  E !

10. 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 E shiftfit E’ shift of E axis EE (only red signal) I E E’ interpolated points (splines) from E to E’ extrapolated points (only red signal) integration with the rest of signals using a common E axis

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

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

13. shiftcalc shiftfit Analyst 133 (2008) 112

14. Example of shiftfit application: Zn 2+ - glycine system Analyst 133 (2008) 112 lof. 15.7 % lof. 6.7 %  1,  2 values 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,  E 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. Example of pHfit application: Cd 2+ - PC 2 1:4 at different pH values: Analyst 135 (2010) 1653 experimental matrix reproduced matrix reference signals corrected matrix svd exp. cor. concen- trations  E vs. pH lof. 16.7 % GlyCysγ-Glu PC n

18. 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 a c I E related to b related to d

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

20. Example of GPA application: PC 5 at different pH values Anal.Chim.Acta 689 (2011) 198 experimental matrix reproduced matrix svd error matrix concentrations vs. pH from currents from areas  E vs. pH w 1/2 vs. pH (width) lof. 5.4 %

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

22. MethodAdvantagesDrawbacks MCR-ALS The best method for the analysis of bilinear voltammetric data. (or very close to bilinearity) Non bilinear data require an unrealistic high number of components to minimise the lof. shiftfit Very useful to correct signals of any shape which move along the potential axis as a previous step for MCR-ALS analysis. When close signals move together the program can ‘confuse’ them. Signals which change their width and/or shape along the matrix produce a too high lof. pHfit Can deal with a complex set of signals moving simultaneously due to sigmoid/linear fitting of potential shifts. It can be wether a pretreatment or an alternative to MCR-ALS. Especially useful for pH titrations, it can be extended to other kind of experiments. Signals which change their width and/or shape along the matrix produce a too high lof. GPA Very useful for signals which change their width and/or shape along the matrix. In these cases it is a valuable alternative to MCR-ALS. The fitting is made row by row, so there are very scarce connections between the fitting of individual voltammograms.

23. Present and future trends: (asymmetric logistic function) 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 Free download and additional information about the programs at: http://www.ub.edu/dqaelc/programes_eng.html