Anal. Chem. 2006, 78, 8165-8167
Correspondence
Method To Improve Linearity of Diffuse Reflection Mid-Infrared Spectroscopy Lacey A. Averett and Peter R. Griffiths*
Department of Chemistry, University of Idaho, Moscow, Idaho 83844-2343 Diffuse reflection (DR) spectrometry using mid-infrared Fourier transform spectrometers has become a popular technique for measuring the spectra of powdered materials.1-3 In most midinfrared DR measurements, samples consist of a mixture of the analyte and a nonabsorbing matrix. Dilution of the sample is necessary because the effect of front-surface reflection distorts the DR spectrum excessively for neat samples. The reflectance of a sample, R∞, is obtained by calculating the ratio of the singlebeam spectrum of an “infinitely thick” sample to that of the matrix, which is commonly powdered KCl or KBr. (The term “infinite thickness” implies that the spectrum does not change when the thickness of the sample is increased.) Spectra are then typically converted to the Kubelka-Munk function, f (R∞), by the operation 2
f (R∞) ) (1 - R∞) /2R∞
(1)
Kubelka-Munk theory4,5 shows that f (R∞) is proportional to the ratio of the absorption coefficient, k, to the scattering coefficient, s, of the sample. Thus, for binary mixtures of an absorbing analyte mixed with a nonabsorbing matrix, a plot of f (R∞) versus the concentration of the analyte, C, should be linear. In practice, several factors can lead to nonlinearity of these plots. These include absorption by the matrix, the effect of specular (Fresnel) reflection from the front surface of the sample,6 baseline offsets,7 and variations in the packing of the sample.8 It should also be noted that the preparation of homogeneous mixtures of powdered components can be very difficult,9 so the standard deviation of plots of f (R∞) versus C (known as Kubelka* Corresponding author. E-mail:
[email protected]. (1) Griffiths, P. R.; Fuller, M. P. Anal. Chem. 1978, 50, 1906-1910. (2) Chalmers, J. M.; Mackenzie, M. W. Solid Sampling Techniques In Advances in Applied Fourier Transform Infrared Spectroscopy; Mackenzie, M. W., Ed.; John Wiley and Sons: Chichester, UK, 1988; Chapter 4, pp 105-188. (3) Blitz, J. P. Diffuse Reflectance Spectroscopy. In Modern Techniques in Applied Molecular Spectroscopy; Mirabella, F. M., Ed.; Wiley-Interscience: New York, 1998; Chapter 5, pp 185-220. (4) Kubelka, P.; Munk, F. Z. Tech. Phys. 1931, 12, 593. (5) Kubelka, P. J. Opt. Soc. Am. 1948, 38, 448. (6) Griffiths, P. R.; Fuller, M. P. Mid Infrared Spectrometry of Powdered Samples. In Advances in Infrared and Raman Spectroscopy; Clark, R. J. H., Hester, R. E., Eds.; Heyden Publishing Co.: London, 1982; Vol. 9, Chapter 2, pp 63-129. (7) Griffiths, P. R. J. Near Infrared Spectrosc. 1996, 3, 6062. (8) Yeboah, S. A.; Wang, S-H; Griffiths, P. R. Appl. Spectrosc. 1984, 38, 486491. (9) Hamadeh, I. M.; Yeboah, S. A.; Trumbell;, K. A.; Griffiths, P. R. Appl. Spectrosc. 1984, 38, 486-491. 10.1021/ac061627o CCC: $33.50 Published on Web 10/28/2006
© 2006 American Chemical Society
Figure 1. Baseline-corrected Kubelka-Munk intensities calculated with the baseline in the reflectance spectrum at (from top to bottom) 0.1, 0.126, 0.163, 0.198, 0.281, 0.398, 0.598, and 1 plotted vs the true value of the Kubelka-Munk function.
Munk plots) can be quite large. In this correspondence, we examine the effect of baseline correction on the linearity of Kubelka-Munk plots for samples prepared in nonabsorbing matrixes at concentrations well below the point at which Fresnel reflection should lead to nonlinearity. Binary mixtures of caffeine and KCl were used as a model system. Baseline variations in DR spectrometry can be produced by simply raising or lowering the sample10 and differences in the morphology of the upper sample layers. Thus, nonzero baselines are very commonly encountered in DR spectrometry. Quantitative errors due to baseline offsets in DR spectra are actually caused by the form of the Kubelka-Munk function. After conversion of the R∞ spectrum to f (R∞), the effect of a baseline offset is to lead to changes in the band intensity (as measured from the band maximum to the baseline), whereas this is not the case if spectra are plotted as log(1/R∞).7 The magnitude of this effect is shown in Figure 1, which shows how the baseline-corrected value of f (R∞) varies with the true value of f (R∞) for baselines from a reflectance value of 1.0 (i.e., the true spectrum) to a reflectance of 0.1. It can be seen that the slope increases dramatically as the baseline offset increases. A plot of the slope of these curves against the value of the baseline in units of R∞ is shown in Figure 2. Note that this curve is approximately linear for 0.7 < R∞ < 1.0, but for more severe baseline offsets, there is a considerable nonlinearity. (10) Murthy, R. S. S.; Blitz, J. P.; Leyden D. E. Anal. Chem. 1986, 58, 31673172.
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Figure 2. Slope of the lines shown in Figure 1 for 1 < f (R∞) < 8 plotted as a function of the baseline reflectance. When R∞ ) 1, the slope of the curve is 1.0.
Recently, Samuels et al.11 reported a technique to improve the linearity of Kubelka-Munk plots caused by relatively small baseline offsets. For a given sample, j, they observed a linear relation between f (R∞) and the baseline position, Bj, so that
f (R∞,ν)j ) f0(R∞,ν)j + A(ν)Bj
(2)
where f0(R∞,ν)j is hypothesized to be the “true” peak height value for the spectral band and A(ν) is the slope of the line at each wavenumber, ν, as a function of multiple measurements presenting varying levels of baseline offset Bj. The effect of baseline offset improved the y-intercept of Kubelka-Munk plots at the expense of linearity. From this intensity, Samuels et al. derived a proposed correction f′0(ν)j to the observed spectral band intensity
f′0(R∞,ν)j ) f (R∞,ν)j - A(ν)Bj
(3)
They stated that if the linear relationship in eq 2 holds true, then the statistical uncertainty in the residual error in f (R∞) values for replicate measurements of DR spectra should be reduced upon applying the proposed correction in eq 3. They proposed that the mean square error as a function of ν for n measurements
1
n
∑[f (R ,ν) - f (R ,ν)]
n j)1
∞
j
∞
2
Figure 3. (A) Measured DR spectra of 10 samples of 10 mg/g caffeine in KCl; (B) result of baseline correction of the spectra shown in (A); (C) result of applying MSC to the spectra shown in (B).
(4)
should be computed to determine the error in their corrected results. The data shown subsequently in their paper validated this approach. This is a rather complex approach to a relatively simple problem. A far simpler and more general approach involves correcting the original R∞ spectrum by locating the maximum datum in a region where absorption by the analyte is minimal and dividing the entire spectrum by this value. The spectrum is then converted to f (R∞). This approach is illustrated using one of the data sets that were originally used by Samuels et al. (These data were originally measured in our laboratory.) Mixtures of caffeine and KCl at concentrations 5, 10, 20, and 50 mg/g were (11) Samuels, A. C; Zhu, C.; Williams, B. R; Ben-David, A.; Miles, R. W. Jr.; Hulet, M. Anal. Chem. 2006, 78, 408-415.
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prepared gravimetrically, and DR spectra of replicate samples were acquired, with 10 spectra being measured for each concentration. Fresh samples were loaded for each measurement. The original f (R∞) spectra for the 19 samples of caffeine in KCl at a concentration of 10 mg/g are shown in Figure 3A. Single-point baseline correction was carried out by dividing each R∞ spectrum by its maximum value between 4000 and 3993 cm-1, where caffeine has minimal absorption. After baseline correction, the spectrum was converted to f (R∞). The data after baseline correction are shown in Figure 3B. There are still some differences between the replicate spectra shown in Figure 3B that can be ascribed either to changes in the scattering coefficient caused by differences in the way that the samples were loaded or to differences in the concentration of caffeine from one sample to the next because the bulk sample from which each individual aliquot was taken was not completely
Table 1. R2 Values from Kubelka-Munk versus Concentration Plots with Various Corrections to the Caffeine Spectra.
Figure 4. Values of f (R∞) for four representative bands of caffeine vs the concentration of each sample in KCl after baseline correction and MSC.
homogeneous. Both causes should lead to multiplicative errors and should, therefore, be eliminated by the application of multiplicative scatter correction (MSC).12-14 MSC was originally developed to remove errors caused by the multiplicative effect of variations of s from sample to sample in near-infrared DR spectroscopy but will in fact compensate for any multiplicative error. (Thus, several recent books on chemometrics call MSC multiplicative signal correction rather than multiplicative scatter correction.) The effect of applying MSC to the spectra shown in Figure 3B is shown in Figure 3C; the spectra are now virtually superimposable. Kubelka-Munk plots of the individual values of the data after baseline correction and MSC for the four caffeine bands measured by Samuels et al. are shown in Figure 4. The values of the correlation coefficient of the plot for each band (shown on the figure) are essentially identical to the corresponding values reported by Samuels et al. after application of their method of baseline correction and MSC. We would note that this approach to baseline correction will allow valid baseline correction even in (12) Geladi, P.; MacDougall, D.; Martens, H. Appl. Spectrosc. 1985, 39, 491500. (13) Masserschmidt, I.; Cuelbas, C. J.; Poppi, R. J.; De Andrade, J. C.; De, Abreu, C. A.; Davanzo, C. U. J. Chemom. 1999, 13, 265-273. (14) Isaksson, T.; Naes, T. Appl. Spectrosc. 1988, 42, 1273-1284. (15) Abrahamsson, C.; Svensson, T.; Svanberg, S.; Andersson-Engels, S.; Johansson, J.; Folestad, S. Opt. Express 2004, 12, 4103-4109. (16) Johansson, J.; Folestad, S.; Josefson, M.; Sparen, A.; Abrahamsson, C.; Andersson-Engels, S.; Svanberg, S. Appl. Spectrosc. 2002, 56, 725-731. (17) Abrahamsson, C.; Johansson, J.; Andersson-Engels, S.; Svanberg, S.; Folestad, S. Anal. Chem. 2005, 77, 1055-1059.
method
3112.6 cm-1
1598.7 cm-1
1189.9 cm-1
860.1 cm-1
uncorrected MSC, Samuels et al. baseline corrected KMC, Samuels et al. baseline corrected and MSC
0.6460 0.95741 0.9244 0.9746 0.9624
0.7002 0.98492 0.8671 0.9844 0.9926
0.7017 0.98775 0.8470 0.9802 0.9955
0.6515 0.98182 0.7564 0.9639 0.9782
the nonlinear region of Figure 2, whereas the protocol described by Samuels et al. should only give good performance where this curve is linear, i.e., when 0.7 < R∞ < 1.0 (vide supra). Since there is a considerable spread in the data in Figure 4, especially when C ) 50 mg/g, and MSC should remove the effect of all multiplicative sources of error, one may ask why the application of MSC does not completely remove the effect of sample-to-sample variations in s and C, both of which should lead to multiplicative errors. We believe that the cause of the residual error is the difference in the path length of diffusely reflected photons from sample to sample caused by small differences in concentration, particle size, or scattering coefficient between each sample. Photon time-of-flight (sometimes known as photon migration) measurements in the near-infrared region show that the effective path length traveled by photons that either penetrate the sample and then re-emerge from the top of the sample or are transmitted through a scattering medium is strongly dependent on the absorption coefficient k.15-17 The effect of differing path lengths will almost certainly lead to variations in f (R∞) of the type seen for the 5% concentration values in Figure 4. The values of the correlation coefficient, R2, are listed in Table 1 for the uncorrected spectra (first row), for the intermediate spectra corrected using the methods of Samuels et al. (second row) and us (third row), and for the spectra that had been fully corrected using Samuels’ method (fourth row) and our method (fifth row). It can be seen that the correlation coefficients for spectra corrected using the method reported by Samuels et al.11 and the approach reported in this paper give statistically identical values of R2. Thus, both baseline correction and multiplicative scatter correction are needed if linear plots are to be obtained using the method reported in this note. Received for review August 31, 2006. Accepted October 16, 2006. AC061627O
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