Selection of the Model for Global Analysis
The first step in analyzing time-dependent spectral data via global
analysis is identifying the number of components, or factors,
contributing to the data. For drying oils, IR peaks associated with each
of the different olefin configurations found during autoxidation (Fig.
3) have been assigned using model compounds (Muizebelt et al. 1994;
Hubert et al. 1997). Several of these peaks are visible in the
time-dependent FT-IR spectra of linseed oil during autoxidation by 1.36
mM cobalt(II) hexafluoroacetonate hydrate (Co(hfac)2) as
shown in Fig. 4. Specifically, the peaks at 3010 and 723
cm-1 are assigned to cis -alkene groups
initially present in the oil (Fig. 3, A ); the peak at
985 cm-1 is assigned to cis-trans conjugated
alkene groups formed after initial H-atom abstraction (Fig. 3,B ); and the peak at 970 cm-1 is assigned to
isolated trans -alkene groups formed via chain propagation and
termination (Fig. 3, C ) (Heredia-Guerrero et al. 2014).
Model-free singular value decomposition (SVD) and evolving factor
analysis (EFA) confirm the presence of two to four distinct factors
contributing to the IR spectra (Fig. S3). Thus, it should be possible to
include at least these three components in the kinetic model. However,
each of these IR peaks shows some overlap with other functional groups
present in the linseed oil, with overlap particularly severe in the
600-1450 cm-1 fingerprint region. Depending on the
relative concentrations of each of the different alkene group
configurations during autoxidation, this overlap may cause difficulty in
uniquely distinguishing one or more of the factors in the kinetic model.
To identify the kinetic model most appropriate to the FT-IR data, models
containing two, three, and four components and a mix of first and second
order rate laws were tested on the FT-IR spectra collected during the
autoxidation of linseed oil by 1.36, 2.04 and 2.72 mM
Co(hfac)2.
In line with the currently accepted mechanism, all models including
second order rate laws provided very poor fitting to the data and were
not further examined. Modeling the 1.36 mM Co(hfac)2data using two components, i.e. A B , provided a
reasonable fit, with the standard deviation of the residuals
(σr) = 1.0*10-3 and the sum-of-squares
(ssq) = 1.0*10-1. The largest errors in this
two-component model appeared in the carbonyl stretching region near 1790
cm-1 and in the fingerprint region (Fig. 5a). The
three-component, i.e. A B C , model provided a
significantly improved fit to this data set, with σr =
6.0*10-4 and ssq = 2.9*10-2, and
greatly reduced error in both the carbonyl stretching and fingerprint
regions (Fig. 5b). Furthermore, the predicted species associated
spectrum (SAS) for species B using this model (Fig. 6b) is
distinct, does not appear in the two-component model, and has a peak at
985 cm-1 corresponding to the cis-transconjugated intermediate proposed in the currently accepted mechanism
(Fig. 3). Inclusion of a fourth component in the model, i.e. AB C D , did not significantly improve the fit
to this data set, with σr = 4.0*10-4and ssq = 1.5*10-2, and exhibited comparable errors in
the carbonyl stretching and fingerprint regions. More significantly,
inclusion of a fourth component led to features with negative
absorptivity constants, a strong indicator of overfitting (Fig. 6c).
These behaviors were also seen in the 2.04 and 2.72 mM data sets (Fig.
S4). Based on this model testing, the three component AB C model was determined to be the most appropriate
for fitting the FT-IR spectra of linseed oil during autoxidation.
For each data set, the values of k1 and
k2 in the A B C would begin
to converge on the same value, i.e. k1 ≈
k2. In order to determine whether k1was, in reality, equal to k2 within experimental error,
we further tested a three-component model with the constraint
k1 = k2 using the 1.36 mM
Co(hfac)2 data set. In the model where
k1 and k2 are optimized freely, values
of 6.4*10-3 and 7.4*10-3min-1 are found for k1 and
k2, respectively. When the constraint k1= k2 was added, a value of 6.9*10-3min-1 was found, with no change in the values of
σr or ssq. This suggests that allowing
k1 and k2 to vary independently does not
improve the quality of the fit. Furthermore, the residual spectra and
SAS generated are effectively identical between the two models, with the
only difference being almost imperceptible changes in the molar
absorptivities predicted for species B (Figs. 7c and 7d).
Similar results were observed when these models were applied to the 2.04
and 2.72 mM data sets. Based on these observations, the three-component
model with the constraint k1 = k2 was
selected to fit the linseed oil autoxidation data presented in this
manuscript. SAS and concentration profiles for all samples studied can
be found in the supporting information (Figs. S5 – S13).
Rates calculated from global analysis (kfit) were
compared with the rate of decrease in peak area at 3010
cm-1 (k3010). As shown in Table 1,
greater rate constants are observed using global analysis relative to
the single-wavelength model. This is caused by the presence of intensity
at 3010 cm-1 in the SAS of species B andC due to the broad peak centered at 2917
cm-1, indicating that the intensity observed at 3010
cm-1 is not solely attributable to isolatedcis- olefins, and suggesting the single-wavelength method
underestimates the rate of initiation during linseed oil autoxidation.
Another contrast between the rate constants determined by global
analysis and those determined using the 3010 cm-1 peak
area is seen in the drier concentration dependence. The global analysis
model indicates that the maximum rate of autoxidation is found at a
concentration of 2.04 mM for each of the three Co driers tested, with
lower observed rate constants at both higher and lower concentrations.
This is consistent with prior observations that the rate of drying oil
autoxidation is inhibited at high catalyst concentrations. Meanwhile,
the rate constants derived from measuring the peak at 3010
cm-1 show an inconsistent dependence on catalyst
concentration, with the maximum rate observed at 1.36, 2.04 and 2.72 mM
for Co(acac)2, Co(hfac)2 and
Co(oct)2, respectively. Taken together, these
differences in rate constants suggest that global analysis provides a
more robust method for determining the rate of linseed oil autoxidation.