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.