Transcript Document

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
Voltammetric measurements: current vs. potential
I/A
E/V
Power
source
RE
AE
AE
WE
WE
I
WE
AE
RE
V
RE
Intricate relationship between current and bulk concentration of species
Ox. + ne-
Red.
Diffusion layer: 1 mm – 1mm
(affected by the electrochemical process)
• mass transport
Electrode
• difusion / accumulation
• homogeneous reactions
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
Usually, signals have characteristic shapes and a rigorous
theoretical model available (Hard modelling)
Ex: Differential Pulse Polarography, DPP
-80
Applied potencial
tp
E
I = I2 – I1 = I (E+E) – I (E)
Signal:
voltammogram
-60
2
Ep
W1/2
-40
tp ≈ 50 ms
td ≈ 1 s
Ep ≈ 50 mV
1
Ip a c*
-20
½ Ip
0
td
t
-0,4
-0,6
1
 
1
2
1

t p 2 PA 2  PA
 nF E

PA  exp 
 E1 
2
2
 RT

   P
2
A
 
 nF E 
  exp 
2 
 RT
-0,7
-0,8
E
Ep
I DPP   I  n F A c*D 2 
on:
-0,5

 PA  PA2 
However, in the presence of many overlapping signals,
multivariate analysis is required (Soft modelling)
chemical species
I/A
component
single electrochemical process
single electrochemical signal
(usually, a peak)
shape constraints
(Cd2+ + Pb2+ + PC5 system)
E/V
In many cases voltammetric data are bilinear :
I   ci I i
i
... thus bilinear methods like MCR-ALS can be applied
peaks stay
at the
same potential
MCR-ALS
scheme
... but sometimes voltammetric data are non-bilinear :
This can be noticed by:
- movement of signals along
E axis
- changes in signal width or
symmetry
10
I/A
8
-8
x 10
5
- too high number of
components by SVD
6
pH
0
4
-1
(Cd2+
-0.9
-0.8
-0.7
+ PC2 system 1:4)
-0.6
-7
sv
I/A
2
-0.5
E/V
- too high lof by MCR-ALS
7
x 10
And can be due to:
-8
x 10
6
5
5
4
4
3
3
2
2
1
1
0
-1
-0.9
-0.8
-0.7
-0.6
0
0
-0.5
E/V
- fast equilibrium between
electroactive species
- changes in electrochemical
reversibility
- changes in homogeneous
reaction kinetics
5
10
15
component
20
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 E
(potential shifts) values, one for every voltammogram.
I
E
pure signals
(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 E
(potential shifts) values, one for every voltammogram.
pure signals without E would yield
a (bilinear) corrected matrix:
I
I
E
Thus, we have to find 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
integration with
the rest of signals
using a common
E axis
I
shiftfit
(only red
signal)
E
E
extrapolated
points
shift of
E axis
interpolated
points (splines)
from E to E’
(only red
signal)
E
E’
E’
The algorithm is somewhat more involved…
(shiftfit/shiftcalc programs based upon the
Matlab command lsqcurvefit)
Analyst 133 (2008) 112
How does lsqcurvefit works?
estimated E for
every component
experimental
voltammogram
Inside shiftfit, for every voltammogram:
[delta]=lsqcurvefit('shiftcalc',delta0,cv,Iexp,lv,uv,options);
contains all parameters
not to be optimised
optimised E for
every component
1 or 0 to indicate if
the component moves
lower and upper
possible values
of E
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)
shiftcalc
Analyst 133 (2008) 112
shiftfit
Example of shiftfit application: Zn2+- glycine system
Analyst 133 (2008) 112
lof.
15.7 %
lof.
6.7 %
1, 2 values closer to
Literature than those
obtained by MCR-ALS
 nF

F0  exp 
E  
 RT

 1  1[glycine]  2 [glycine]2
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
E  a  b pH
sigmoid
E 
a
d
1  exp b ( pH  c)
(signals can be also immobile
or randomly moving with pH)
Analyst 135 (2010) 1653
The algorithm:
Analyst 135 (2010) 1653
Example of pHfit
application:
Cd2+- PC2 1:4
at different pH
values:
experimental
matrix
Cys
svd
cor.
reproduced
matrix
PCn
γ-Glu
exp.
concentrations
Gly
E vs. pH
lof. 16.7 %
reference
signals
corrected
matrix
Analyst 135 (2010) 1653
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:
I
left side of maximum:

Ileft  c exp  b (E  a)2

right side of maximum:

Iright  c exp  d (E  a)2
c

- 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
related to b
E
related to d
1 
 1
w1 / 2  ln 2 


b
d


Area 
c   1
1 



2  b
d
The algorithm:
Anal.Chim.Acta 689 (2011) 198
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
reproduced
matrix
from areas
error matrix
E vs. pH
w1/2 vs. pH
(width)
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 %
MCR-ALS 2 comp. (reproduced and error matrices)
lof.
4.6 %
GPA 1 comp. (reproduced and error matrices)
Method
MCR-ALS
shiftfit
pHfit
GPA
Advantages
Drawbacks
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.
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.
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.
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.
Present and future trends:
• Application of other parametric functions in programs analogue to GPA
(asymmetric logistic function)
Analyst 136 (2011) 4696
• 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)
Free download and additional information about the programs at:
http://www.ub.edu/dqaelc/programes_eng.html