Effect of Nonuniform Distribution of Absorbing Material cm the Quantitative Measurement of Infrared Band Intensities J. L. KOENIG' Plasfics Department, Research & Development, Division,
b The effect of a wedge-shaped sample on the quantitative measurement of infrared band intensities is discussed and some numerical calculations are reported. Conclusions are drawn about the effect of a wedgeshaped sample on ordinary peak height measurements and band ratio measurements using internal standard bands. For compensation or difference spectra measurements, the band shape distortion caulNed by wedgeshaped sample can introduce appreciable error.
T
us 3 of polymer film samples for quantitative infrared meaqurements of end groups, copolymer composition, stereoregularity, and crystallinity is evident from an examination of the current literature. The inherent difficulties asr,ociated with the preparation of such fdms requires an analysis of their acceptability as samples for quantitative measurement. Film samples obtained by pressing or solvent casting meth3ds are usually uneven. Polymers Rhich cannot be melted, such as polytetrafluoroethylene and polyurethanes ( 1 ) must be microtomed; from this twatment, uneven surfaces result. Voids are often present in the sample films. The uneven distribution of sample obtained in a KBr pellet (6) is another common defect of sample preparation. A perfectly uniform cell or polymer film is often characterized by the appearance of interference fringes in wavelength regions of nonabsorption. These undesirable fringes affect the background so base lines cannot be drawn unambiguously. SO, iinperfect samples are most often used. Rut in all quantitative infraied ineasurements, it is implicitly assumed that a uniform sample is being examined. The simplest way to consider the irregular distribution c f material in the sample beam is to examine the effect of a wedge-shaped samp e along the slit. Variations across the slit are possible, HE INCREASING
Present address, Engineering Division, Case Institute of Technology, Cleveland
6, Ohio.
E . 1.
du font de Nemours &
but because of the narrow slit widths normally used, they mill be neglected. The magnitude of the wedge can usually be determined by measurement, and small variations can be neglected. It is the purpose of this paper to indicate the nature of this effect and to report the results of some numerical calculations.
Procedure. The effect of a wedgeshaped sample is most easily explained by considering a film which covers only half of the spectrometer's entrance slit. If the specific absorbance of the film is zero, the spectrophotometer will record zero. R u t if the absorbance of the film is infinite, the machine, which records the per cent transmittance of the sample as presented to it. will record 50y0 transmittance. A calculation of the absorbance of the sample from the transmittance measurement is obviously wrong. The quantitative relationship between the specific absorbance and the measured absorbance can be derived by considering the simple case of a material in the beam of the spectrometer with a fraction, 2, of the entrance slit covered with a sample of path length, b', and remainder (1 - 7)with sample with path length b" (b" > b f ) . I t is here assumed that the change in thickness across the slit is negligible for the narrow slits of the infrared spectrometer. The instrument is recording the total energy transmitted = SI'
+ (1 - z)l" = I & -
(1)
where a, is the apparent absorptivity of the material as presented to the instrument, and b is apparent path length observed by spectrometer. Simple substitution results in the following equation relating the true absorptivity and the apparent absorptivity for a two part sample e-aac
[zb'
re-
+ (1 - z ) b " ] = arcb'
+
(1 - r)e- aib"c ( 2 ) When the fraction of sample with path length, b", has a specific absorbance of zero, this equation reduces to the form derived by Jones (5). The equation for a continuous uniform change in thickness from b' to 6" can be obtained by integrating over the length of the slit. The path of any point x on the
Wilmingfon, Del.
+
slit will be b' z(b" equation becomes: e-
1
4-z(b" -
aacJ[b'
- b'), so the above
b ' ) ] dt =
1
$e-
(ItC
[b'
z(b"
-
b')l
d2
(3)
After integration this becomes: e-
EXPERIMENTAL
I
Co.,
b" f b'
aac
(--z-)
=
The wedge effect can be defined in terms of a single parameter by the equation 6"
=
6=-
b' (1 b"
+ 6)
- b'
(5)
b'
The value of 8 can be approximated by measurement of film thickness and is better than an average thickness. Then, the equation may be written in the form ,-aacb'
(1
+ 6/2)
=
RESULTS
This equation has been evaluated numerically on the IBM 650 computer. Part of the results are tabulated in Table I. The discrepancy between the apparent absorbance (defined: A , = log,, I,/I = a,cb) and the specific absorbance ( A , = loglo I,/I = atcb')! for a given wedge increases as the absorbance increases. As the wedge becomes greater, the discrepancy increases accordingly for the same observed absorbance. Thus, absorption bands will be distorted and flattened a t the top for uneven samples. From these numerical results, several useful conclusions can be drawn about ordinary peak height measurements (4, band ratio measurements with internal standard bands (2, 7), compensation or difference spectra measurements (8). For ordinary peak height measurements in the usual range of absorbance, the effect of a wedge on the absorbance VOL. 36, NO. 6, M A Y 1964
1045
Table
I.
Variation
of Measured Absorbance with Real Absorbance as Function of Wedge Parameter
True absorbance
0.01
0.21
1.0000
1.0000 0.9000
0.9962 0.8969 0.7976 0.6981 0.5986 0.4990 0.3994 0.2997 0.1998 0,0999
0.9000 0.8000
0. 7000
FRLWENC"
0.6000 0.5000 0.4000 0.3000 0.2000 0.1000
0.8000
0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000
Wedge parameter (6) 0.41 0.61 Measured absorbance 0.9867 0.9731 0,8892 0.8781 0.7915 0.7827 0.6935 0.6867 0.5952 0.5902 0,4967 0.4932 0.3979 0,3956 0.2988 0.2975 0.1995 0.1989 0.0998 0.0997
0.81
1.01
0.9564 0.8645 0.7718 0.6784 0.5840 0.4889 0.3929 0.2960 0.1982 0.0995
0.9377 0.8491 0.7595 0.6688 0.5770 0.4839 0.3897 0.2942 0.1974 0.0993
Figure 1. Effect of wedge-shaped sample on difference spectra
is very small. For example, for an absorbance of 0.300 and a wedge where the thickness doubles along the dit, the measured absorbance is 0.294 or a change of 2% which is within the normal precision of the measurement. If integrated band intensities are used, similar deviations occur. However, when the specific absorbance is higher, the effect increases appreciably, For cases where a very high absorptivity exists, or very high concentration of absorbing material requires a very thin film or liquid cell, the wedge effect will result in a departure from Beer's law since the measured absorbance will not be directly proportional to concentration. It is precisely for these cases that the possibility of obtaining a wedge is the greatest, so extreme care must be taken if accurate measurements of specific absorbances are to be obtained. The obvious way to measure the thickness of a film (or quantity of material in a KBr pellet or Nujol mull) is to use the intensity of a band of known origin which can be correlated with the thickness and density of a polymer film (or the concentration of material in the pellet or mull). However, this does not eliminate the wedge effect, unless the internal thickness band has the same intensity as the analytical band. In this case, the wedge error cancels. However, this is a trivial case since such measurements dictate that the measured band is not related directly to the internal thickness band. Whenever the internal thickness band is weak, as when an overtone is selected, the ratio will contain a measurable error due to the wedge effect for nonuniform samples. This has been observed experimentally for machined nylon samples ( 3 ) . Our results indicate, for a wedge where the path length doubles along the slit, that ratio can be 0.8/0.1 = 8, but that measured ratio will be 0.7595/0.09935 =
1046
ANALYTICAL CHEMISTRY
7.644 or 5% low. The case of a very strong internal thickness band leads to similar difficulties but the ratio will be too high by a similar amount. It is obvious that an extreme case has been chosen and the error is large. Under normal circumstances this type of measurement is the least affected by a wedge-shaped sample. If the difference in the intensity of the analytical and internal thickness bands is not large, the effect on the results for a high or low ratio will be small and probably below the normal precision. However, if very thin films are used, if the films possess large numbers of voids, or if strong absorbances are measured, careful attention to the wedge effect n7ill eliminate wide variations in results. A difference spectrum is much more precise than an ordinary measurement when an interfering absorbance is to be eliminated. When an uneven sample is present in both beams, the effect is complicated. When the wedginess of the samples is the same in both beams, the effect cancels. However, when there is a difference in the wedginess of the two samples, the interfering absorbance cannot be completely removed. This is demonstrated in Figure 1. For convenience, the band is assumed to have a Gaussian shape. Band 9 is the absorption band for a flat sample with an absorbance of 1.0. Band B is the distorted band corresponding to the measured absorbance of Band A for a wedge of double thickness along the slit, Band C is a Gaussian curve with an absorbance of 0.937 a t the peak frequency rvhich should be used to compensate for the band B. The shaded area is the absorbance which is not removed because of the differences in band shape. The lower curve is the measured absorbance curve for a difference spectrum with band C in the reference beam and band B in the sample beam. If the analytical band lies a t a frequency which places it under
the wings of the residual band, this residual absorbance will be added to the true absorbance of the analytic band. For weak absorption bands, this could account for a large portion of the measured absorbances. CONCLUSION
The wedge effect is serious only when very strong absorbances are measured or when very thin films or liquid cells are used where the amount of wedge may be appreciable. The magnitude of the wedge can sometimes be determined by scanning different portions of samples while masking the remainder from the beam. For cells, the quality of the interference fringes obtained from the empty cell is an indication of the shape of the cell. The internal thickness band method incurs least error due to the wedge effect. Compensation methods suffer more from this effect and careful attention to the quality of films is necessary for accurate results. With compensation techniques, the data from different samples should not be averaged, but should be fitted to a straight line to minimize the wedge effect. LITERATURE CITED
(1) Corish, P. J., A N A L . CHEM. 31, 1298 ilS.SRi ,-~~",. (2) Ibid., 33, 1798 (1961). (3) Doskocilova, D., Schneider, B., Sebenda, J., Collection Czech. Chem. Commun. 27,1760 (1962). (4) Heigl, J. J., Bell, B. F., White, J. Y., ANAL.CHEM.19, 293 (1947). (5) Jones, R. N., J . Am. Chem. SOC.74, 2681 (1952). (6) Magnasco, Ti., Rossi, >I., J . Polymer Sci. 62, 5172 (1962). ( 7 ) O'Connor, R. T., DuPrB, E. F., McCall, E. R.. .4r;a~. CHEU.29, 995 (1957). (5) Willbourn, A. H., J. Polymer Sci. 34, 569 (1959). RECEIVED for review December 13, 19G3. Accepted February 10, 1964.
