Application of New Least-squares Methods for the Quantitative Infrared Analysis of Multicomponent Samples

David M Haaland and Robert G Easterling

Sandia National Labratories

Applied Spectroscopy Volume 36, Number 6, February 1982

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Improvements have been made in previous least-squares regression analyses of infrared spectra for the quantitative estimation of concentrations of multicomponent mixtures. Spectral baselines are fitted by least-squares methods, and overlapping spectral features are accounted for in the fitting procedure. Selection of peaks above a threshold value reduces computation time and data storage requirements. Four weighted least-squares methods incorporating different baseline assumptions were investigated using FT-IR spectra of the three pure xylene isomers and their mixtures. By fitting only regions of the spectra that follow Beer’s Law, accurate results can be obtained using three of the fitting methods even when baselines are not corrected to zero. Accurate results can also be obtained using one of the fits even in the presence of Beer’s Law deviations. This is a consequence of pooling the weighted results for each spectral peak such that the greatest weighting is automatically given to those peaks that adhere to Beer’s Law. It has been shown with the xylene spectra that semiquantitative results can be obtained even when all the major components are not known or when expected components are not present. This improvement over previous methods greatly expands the utility of quantitative least-squares analyses.

Introduction

Quantitative infrared spectroscopy has gained in popularity with the capability of obtaining digitized infrared spectra. Antoon et al.¹ have applied linear least-squares regression methods to the infrared spectra of multicomponent mixtures in order to analyze quantitatively the composition of the individual components. Assuming a linear relationship between concentration and absorbance (i.e., assuming Beer’s Law is obeyed), these techniques have been successful in the quantitative analysis of multicomponent mixtures even in those cases where there is complete overlap of the infrared spectral features. The inclusion of all the data in the spectral region of interest also significantly improves the precision and accuracy of the results. The methods of Antoon et al.¹ have been applied with success in the quantitative analysis of polymer components and the mineral composition present in coal.²

Earlier, we presented least-squares methods for improving the sensitivity of the quantitative analyses of the infrared spectra in regions of the spectrum where only single components were present.³ We were able to improve the sensitivity by simultaneous least-squares fits of the spectra and the baselines. Alternatively, a least-squares derivative fit of the spectra was used to correct for slow nonlinear variations in the baseline. These methods were shown to improve the detection of trace gases by factors of 5 to 7 for gas molecules of low molecular weight and allowed detection even in those cases where the signal was less than the noise. It is useful to combine our earlier methods with those of Antoon et al.¹ so that automatic spectral baseline corrections can be incorporated into the least-squares regression analysis of multicomponent mixtures with overlapping spectral peaks. This combined method of analysis would improve sensi­tivity and eliminate the more subjective baseline corrections required for the sample and reference spectra. Base­line corrections for quantitative analysis are especially difficult and subjective in those cases where there is scattering by the sample or where significant spectral overlap occurs. Incorrect baseline corrections are equivalent to a breakdown in Beer’s Law. Therefore, accurate baseline corrections are of fundamental importance. This paper outlines the extension of our earlier least-squares methods to the case of multicomponent mixtures with overlapping spectral features and its application to artificial and real mixtures.

I.   Experimental

A Nicolet 7199 Fourier transform infrared (FT-IR) spectrometer was used with a liquid nitrogen cooled HgCd-Te detector with a range from 400 to 5000 cm^-1. Interferograms were collected to yield ~2 cm^-1 resolution after transforming the data with triangular apodization. In order to compare results with those of Antoon et al.¹ mixtures of the three xylene isomers (dimethylbenzenes) were used to evaluate the least-squares analyses. The xylenes used were chromatographic standards from Poly Science Corp. with isomeric purity of ≥99.5%. The reference spectra were obtained from the same 15-μm path length liquid cell to assure constant path length. Two hundred fifty-six interferograms were signal averaged in each case. The sample consisted of an accurately weighed (±0.0001 g) mixture of nearly equal weights (~0.15 g) of the three xylenes. Again the same liquid cell was used to eliminate path length variations. The spectrometer was well purged with dry N₂ gas to eliminate H₂O and CO₂ interferences.

The least-squares analysis was applied both to artificial and real xylene mixtures. The artificial mixtures were created by digitally adding known fractions of each xylene reference. Noise at various levels was then added to complete the artificial sample spectrum. The noise was generated by taking two separate single beam spectra of one scan with no sample in the IR beam. These were then ratioed to create a noise spectrum with the normal instrumental noise characteristics. Greater amounts of noise were created by multiplying this noise spectrum by constant factors. These artificial spectra assured that Beer’s Law was followed over the entire spectral region since they were created from a linear combination of the reference spectra. The spectra of the real mixtures were used to determine the effects of possible non-Beer’s Law behavior on the least-squares analysis. To determine the effectiveness of least-squares baseline corrections, the baselines of the samples were either left unchanged or corrected to zero with the interactive Nicolet software before applying the least-squares program to the data.

II. Theory

Beer’s Law is used as the basis for relating the concentration (c) of an absorbing species to its infrared absorbance (A) at each specified energy. That is,

A = abc   (1)

where a is the absorptivity and b is the path length. Beer’s Law generally requires that the resolution of the spectrometer is sufficiently high to avoid significant instrumental broadening of the spectral bands being measured.⁴ Anderson and Griffiths⁵ also show that the instrumental line shape can often affect the measured peak absorption, and they recommend the application of Beer’s Law only when A ≤ 0.7. Eq. (1) also requires that each individual component in a mixture is not affected by the presence of other components (i.e., non-interacting components). As has been shown previously¹ and confirmed in this study, the absorptions due to certain vibrations are often not influenced by the presence of component interactions that affect other vibrations of the same species. These unaffected vibrations will therefore follow Beer’s Law.

Least-squares regression analyses are specifically developed here for samples of multicomponent mixtures with overlapping spectral features. Each analysis uses all the spectral data above a selected absorbance threshold in the spectral region of interest. The least-squares analysis includes a fit of the spectral baselines as well as a quantitative determination of each component of the sample. Since the sample spectrum is fitted to a least squares linear combination of pure reference spectra, no assumptions about spectral line shapes are required.  As before,³ we have developed and tested four different least-squares fitting procedures. The differences between the four are in the assumptions made about the baselines of the spectra. The assumptions are: (I) the baseline is zero, i.e., the case developed previously by Antoon et al.¹ with no baseline fit; (II) the baseline is linear across the spectral region to be fit; (III) the baseline is linear over each spectral peak in the fit; and (IV) there is a negligible baseline shift between successive data points in the fit. This last analysis is equivalent to a least-squares fit between first derivatives of the sample and reference spectra. In general, method I will require a correction of the baseline to zero absorbance before curve fitting since even reflectance losses will result in nonzero baselines. If the baseline assumptions are valid in methods II to IV for both the sample and reference spectra, then no preliminary baseline corrections are necessary for either the sample or reference spectra. However, with sloping baselines it may be desirable to baseline correct the reference spectra prior to curve fitting since this aids in the proper selection of peaks for the fitting program. In general, the baselines of pure reference spectra with high signal-to-noise ratios (S/N) are readily corrected to near zero by the software accompanying most commercial computerized infrared spectrometers.

The appropriate mathematical equations describing the four methods of least-squares analysis and the assumptions about the baselines are given in Table I. A_i^s represents the absorbance of the sample at frequency i while A_ij^r represents the absorbance of the jth reference at the same frequency. The parameters k_j are the ratios of the concentration of the jth component in the sample, c_j^s, to the concentration of the jth reference, cò. (If path length variations are present, k_j represents the ratio of the path length concentration product (bc) for the sample and reference). The k_j terms are therefore the scaling parameters which can be used directly in spectral subtractions of the references from the samples. The equation for method I in Table I simply represents the linear combination of reference spectra which make up the composite sample spectra as expected from Beer’s Law. The e_i term is the random error present at frequency i. The random error is assumed to be normally distributed with an expectation of zero and a variance proportional to T^-2 as discussed earlier. A term that is linear in frequency (ie, a + bv_i) has been added in method II in order to fit a linear baseline over the desired spectral region. Method III has the same linear baseline fit except that a separate linear baseline is fitted for each spectral peak. A different set of k_j is estimated from each peak and the final k_j‘s are determined by pooling the individual k_j‘s weighted inversely by the estimated variance calculated for each component in each peak. Method IV, which fits the difference between successive data points at constant frequency separation, is equivalent to a least-squares fit of the first derivatives of the reference and sample spectra.

Table II presents the same equations in matrix form. The details of the computational methods for each fit are given in the Appendix. In particular, the least-squares solution for method I in matrix form is

where Ar and As are the matrices representing the reference and sample absorbances, the prime indicates the transpose of the matrix, and the Z matrix is the n x n diagonal matrix of weights (i.e., zü = Ti2 = 10^-2Ais, where the T₁ term is the transmittance of the sample at frequency i). If C_r is the vector matrix of known reference concentrations, then the concentration of each component in the sample is calculated by

The details of calculation of the variance of k, the error variance, σ², and the standard error of the estimated concentration; SE(c_j^s), are given in the Appendix. For large n, to a close approximation a 95% confidence interval on the true concentration in the sample is given by c_j^s ± 1.96 SE(c_j^s).