Comparison of Continuum Models in Quantitative Diffuse Reflectance Spectrometry Harry G. Hecht Depatiment of Chemistry, South Dakota State University, Brookings, S.D. 57006
Model systems Containing soluble dyes In scatterlng medla are considered in thls article dealing with quantitatlve diffuse reflectance spectroscopy In highly absorbing medla. Several continuum models are used to interpret the data, including those of Kubelka and Munk, Pltts, and Rorenberg, as well as exact solutions to the radlatlve transfer equatlon for varlous phase factors. The Pitts’ formula is a two-parameter representation which glves a good fit to the experimental data. It does, however, requlre the use of very anisotropic phase functlona. The large deviations from the Kubelka-Munk and similar formulas are due to the anisotropy of scatter. Thus, the two-parameter formula Is not just an empirical means of fitting the data but has the potential of distinguishing between nonlinear effects and instrumental errors, and thus extending the range of reflectance measurements for quantltative analytlcal purposes.
Numerous diffuse reflectance theories have been proposed, and they may be quite different depending on the interests of the author(s).Astrophysicists have usually been concerned with radiative transfer theory, whereas spectroscopists have tended tloward the use of various layer models. It has become traditional to group the theories into two categories, the continuum models and the statistical models. There are many parallels which can be drawn between the theories, and these have reciently been reviewed (1). In general, it may be stated that the statistical theories require the use of a rather specific model for the medium. By summing contributions to the reflectance from the individual particles or layers, one can derive an expression for the reflectance from which the absorptivity can be determined. The success of this approach depends, of course, on how closely the model corresponds with reality. The continuum theories, on the other hand, are more general. They do not depend on a detailed knowledge of the medium since the scattering and absorbing processes are described by two phenomenological constants. The absolute absorbance cannot be determined using continuum theories. For analytical purposes, this is not a severe limitation, since determinations are usually made from comparative measurements. Reflectance spectrometry is now recognized as a valuable companion technique to transmission spectrometry (2). It is, in fact, the preferred technique in some highly absorbing media, and the only applicable technique for many surface studies, color measurements, and in situ analyses of thin-layer chromatograms. The most widely used method of interpreting diffuse reflectance data on a quantitative basis is the Kubelka-Munk theory (3-5). This theory was advanced in 1931 (6), but, in fact, constitutes a re-discovery of the earlier treatment by Schuster (7). The form given by Kubelka and Munk is convenient for analytical purposes, and this is probably the reason that spectrometrists continue to use it so widely and refer to it as the K-M theory. The K-M function can be applied for quantitative analysis
by reflectance measurements in much the same way that the Beer-Lambert law is applied to transmission measurements (3-5). As is well known, the Beer-Lambert law is a limiting law which is obeyed only in dilute solutions. The K-M function should also be regarded as a limiting law for essentially the same reasons (8). In practice, it is found that the K-M function is valid only over a very limited concentration range (9). The primary reasons for this appear to be two basic assumptions which are not valid. Namely, it is assumed that homogeneous scattering layers are involved in which scatter and absorption are proportional to the layer thickness. Differential equations are then written for these layers, although one would not expect them to be valid in the limit of small differential layer thicknesses (10). Furthermore, as the medium becomes more strongly absorbing, anomalous dispersion effects should play an increasingly significant role (11).These effects would also lead to deviations from the ideal K-M behavior, since they are not included in the theory. The K-M theory represents an approximate solution to the radiative transfer equation for a medium which scatters isotropically (2). The exact solution for this case has been given by Chandrasekhar (12). An exact solution for scatter according to the phase function q ( l + x cos e)is also available (12). wo is defined in terms of the absorption ( a )and scattering (a) coefficients by wo = a / ( a a), and is known as the albedo of single scatter. 8 is the angle of scatter referred to the direction of incidence assuming a radial distribution, and x is a parameter representing the degree of anisotropy. The various solutions have been put into a useful form by Giovanelli (13) and compared with an approximate solution due to Pitts (14). Another useful approximation is that of Rozenberg (15). It appears that some of the other solutions mentioned above might give a more accurate description of the reflectance of strongly absorbing media than the K-M function, and would allow such measurements to be used for quantitative analytical purposes over a more extended concentration range. It is felt that this approach is preferable to the use of the numerous ad hoc relationships which have been proposed in the past to linearize reflectance data simply as an analytical expedient (16), since these experiments have at least the potential of suggesting reasons for departure from the simple K-M theory. It is unlikely that a new relationship can be found which linearizes reflectance data much better than the K-M theory over an extended concentration range, but the simple calculational procedures which the other theories involve cannot be regarded as a serious deterrent in a modern analytical laboratory. Several model systems will be used to quantitatively compare the various continuum reflectance theories in strongly absorbing media. This paper reports the results for systems of one type; namely, those containing a soluble absorbent in a scattering medium.
+
EXPERIMENTAL Two model systems were chosen for the present study: eosin B in magnesia suspensions, and methylene blue in cow’s milk. The samples were prepared by quantitative addition from stock dye solutions to
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Figure 3. Reflectance of eosin B in milk of magnesia suspensionsat the two wavelengths 41 1 nm (0)and 516 nm (0)as a function of the reduced concentration, C The dashed curves are least squares fits using an isotropic phase functlon (Equation 7), and the solid curves are least squares fits using the Pitts formula Flgure 1. Some representative diffuse reflectance spectra for samples of eosin B in milk of magnesia Each sample contained 10 ml of Phillips’.Milk of Magnesia and was made up to a total volume of 20 ml. The molar concentrations of dye for the various curves 1.62 X 4.06 X loh5,1.62 X (from top to bottom)are: 4.06 X 3.25 X lo-“
100
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I
sorption peaks involved are identified by their reflectance maxima a t ca. 411 and 516 nm. The absorption and scatter of the samples are represented by the two constants CY and u, respectively, where
I
u
500
400
C2uz
(2)
600
A(nrn1
Figure 2. Transmission spectrum of a 8.12 X 10-6 M aqueous solution of eosin B recorded with a Beckman Model DB spectrophotometer
suspensionswhich contained a fixed volume of the scattering medium. Commercial Phillips’Milk of Magnesia was used in the former case, and fresh whole cow’s milk in the latter. Reflectance measurements were made using a Beckman Model DK-2A spectrophotometer equipped with an integrating sphere reflectance attachment. Similar suspensions to which no dye had been added were used as reflectance standards. Ten-mm silica cells were used. It was verified that the condition of “infinite layer thickness” was satisfied by noting that the reflectance was not altered by the addition of a white backing. A dark current adjustment was made before each measurement. Periodic checks vs. the reflectance standard showed that instrumental drift was no more than 1%throughout the course of the measurements. Those reflectances which were less than 0.1 R are reported as averages of several scans using an expanded scale.
RESULTS T h e Eosin B in Magnesia Model System. Some typical reflectance curves are shown in Figure 1. By comparison with the transmission spectrum of the dye solution (Figure 2), the peak at shorter wavelength is considerably accentuated. This effect is attributed to increased scatter a t the shorter wavelengths. This same effect is probably responsible for some apparent shifts in the peak positions as well. The two ab1776
= c1u1+
The subscript 1 refers to the magnesia particles and the subscript 2 refers to the dye molecules. It is assumed that a1 = 0 and 02 = 0. The reflectance is expressed in terms of the single independent variable C = Cz/C1,where C1 is a relative concentration determined by the dilution factor. The absolute concentration of scattering particles is not reported here. The measured reflectance values as a function of C are shown in Figure 3 for the two peaks a t 411 nm (0)and 516 nm (0). In Figure 4, the data are plotted according to the K-M function f(R),
zot 300
(Equation 9)
(3) where y = az/al.A least squares fit to the data is represented by the solid lines whose slopes yield the parameters 7411 = 2534 and 7516 = 24168. The ratio y411/y516 = 0.104 is smaller than the corresponding ratio of molar extinction coefficients determined from the solution spectrum, 0.148. This is presumably due to the different magnitude of the scattering constant a t the two wavelengths, as previously mentioned. As might have been anticipated, the K-M theory fails to give a good account of the reflectance for the more intense band, although that for the weaker band is satisfactory. Rozenberg (17) has argued that models of the K-M type are too crude to give anything but a fortuitous agreement with actual reflectance data. Using methods developed by Kuznetsov (18),Rozenberg (15)showed that the reflectance is given by
(4) where t is the multiplicity of scatter, Ci is a parameter describing the polarization of the reflected radiation, and the ait are constants which depend upon the scattering indicatrix. As P = LY/Uincreases, terms of higher multiplicity become less important. Ambartsumian (19) has shown that the mean multiplicity of scatter in a semi-infinite turbid medium is in
ANALYTICAL CHEMISTRY, VOL. 48, NO. 12, OCTOBER 1976
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CXIO‘
’
4
FIgure 4. Kubelka-Munk function for eosin B in milk of magnesia suspensions at the two wavelengths 411 nm (0)and 516 nm (0) as a function of the reduced concentration, C. The lines are least squares fits using Equation 3
w.
fact given by Thus when P 1 Ily, a sufficiently accurate solution is obtained by inclusion of terms up to second or third d.egree. Assuming scatter to be independent of P, the reflectance is then given by R
1 + P-
(5)
Figure 5. Rozenberg function for eosin B in milk of magnesia suspensions at the two wavelengths 41 1 nm (0)and 516 nm (0) as a function of the reduced concentration, C The solid line is a least squares fit to the 516-nm data using Equation 5,and the dashed curve is a least squares fit to the 41 I-nm data using the same fixed value of Q = 16.18
= 5964, and is shown by the dashed curves in Figure 3. The fit is rather poor; the curvature is obviously not correct. The exact solution for the phase factor q ( l +x cos 6) has also been given by Giovanelli ( 1 3 ) ,
R(1) = 1 - H(1)(1- [ ( ~ o / ~ ) (cY c aoi ) ] )
Q
(8)
where
where
c = x(l
is a const.ant characteristic of the scattering medium in the absence of absorption ((3 = PO). Equation 5 can also be derived in terms of a simple layer model (20),in which case a
t andQ=l+(6) r+t r+t a, t, and I’ are the absorption, forward scatter, and reflection constants of the layer, respectively. Figure 5 shows the reflectances a t 411 and 516 nm plotted according to the Rozenberg formula (Equation 5). The solid line is a least squares fit to the 516 nm peak with Q516 = 16.18 and y516 = 17687. The fit is seen to be quite satisfactory in general, although there is an obvious trend in the data at low concentrations which is not represented by the Rozenberg curve. No convergence was obtained in an attempt to fit the 411-nm band. The curvature is so slight that Q is not uniquely determinlad. If‘it is fixed a t the value of the 516-nm band; i.e. at 16.18, a rather poor fit is obtained with 7411 = 4125 (dashed line). An attempt was made to fit the reflectance data using the exact solution to the radiative transfer equation for isotropic scatter in the form given by Giovanelli (13) for our Ro.,D measurements:
p=-
R(1) = 1 - H(1)(1 - o o ) ” 2
(7)
where wo = a / ( a + a) and H(1) is an integral given by Chandrasekhar (12).The least squares fit gives 7411 = 1148 and 7516
- wo)wo[CY1/(2 - w o a o ~ l
and a0 and 011 are moments of the H-integrals. It is generally assumed that 0 5 x I1.An attempt was made to fit the data of Figure 3 using x = 1.The result was somewhat better than with x = 0 (isotropic solution), but still unsatisfactory. This suggests that a fit with x > 1might be better, but the necessary integrals have been tabulated by Chandrasekhar (12) only for x = O a n d x = 1. It was shown by Giovanelli (13) that an approximate solution given by Pitts (14) is in excellent agreement with the exact solution a t x = 0 and x = 1. The Pitts formula is obtained using the Eddington approximation and can be expressed in the form, 3 + (1- w0)x + 3 - oox (-x + 1 + Po*-] WO
R(Po) = 2 v 5
(9)
where
x = (3 - w o x ) ( l - wo) This formula represents a convenient analytic expression for non-integral x values, and makes possible the investigation of systems with very anisotropic scatter without the cumbersome calculation of H-integrals and moments. It will be shown elsewhere (22)that the Pitts approximation is in very good agreement with the exact solution for non-integral x values, including those in range x > 1. A nonlinear least squares fit of Equation 9 to the data is shown by the solid curves of Figure 3. The parameters determined are 7 4 1 1 = 127.5, x4ll = 2.52, 7 5 1 6 = 1278, and x516 =
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Flgure 8. Reflectance of methylene blue in milk suspensions at the wavelengths 660 nm (0), 610 nm (0) and 390 nm (A)as a function of the reduced concentration, C I
I 500
1 I 600A(nm)
l
l
'0°
I
Figure 6. Some representative diffuse reflectance spectra for samples of methylene blue in cow's milk
The dashed curves are least squares fits using an isotropic phase function (Equation 7) and the solid curves are least squares fits using the Pitts' formula (Equation 9)
Each sample contained 25 ml of fresh whole milk to which varying volumes of a stock dye solution were added. The reduced molar concentrations for the various curves (top to bottom) are: 5.35 X 2.67 X 1.07 X 5.35 x 10-4.2.67 x 10-3
/ l
o
0
lot 0
/
l
i 1
1
400
500
1 600
A(n.1
Figure 7. Transmission spectrum of a 6.69 X 10-5 M aqueous solution of methylene blue recorded with a Beckman Model DB spectrophotometer
2.06. The agreement is obviously quite good and the ratio 0.0998 is very nearly the same as that determined by the K-M theory. The Methylene Blue in Cow's Milk Model System. Analogous measurements were made for the second model system. Some typical reflectance curves are shown in Figure 6. Once again a comparison with the solution spectrum of methylene blue (Figure 7 ) reveals some changes in intensity and position which may be attributed to an increase in the scattering constant a t shorter wavelengths. Measurements were made for the peak a t ca. 660 nm, the shoulder at ca. 610 nm, and the small peak a t ca. 390 nm. Reflectance values for these three features are shown in Figure 8 by the symbols 0, 0, and A, respectively. A relative concentration is once again used, since the absolute concentration of scattering particles has not been determined. A plot of the reflectance data according to the K-M theory is shown in Figure 9. The least squares analysis gave the values of 7 6 6 0 = 1911, 7610 = 1490, and 7390 = 8.5. The lines determined by these slopes are given also in Figure 9, and once y411/~516=
1778
Figure 9. Kubelka-Munk function for methylene blue in milk at 660 nm (0), 610 nm (0), and 390 nm (A)as a function of the reduced concentration, C. The iines are least squares fits using Equation 3
again the model is satisfactory for the weak band but gives a very poor fit to the stronger bands. The reflectance data for methylene blue in milk are shown in the Rozenberg form in Figure 10. A nonlinear least pquares fit to these data was convergent only for the 610-nm shoulder for which 7610 = 2188 and Q610 = 3.24. The other bands have curvatures which are not consistent with the Rozenberg formula (Equation 5 ) . A fit of the data with the exact solution for isotropic scatter (Equation 7 ) is shown by the dashed curves of Figure 8. The corresponding y-values are = 1061,7610 = 468, and 7390 = 3.4. It is apparent once again that the curvature of the isotropic solution is too small, and a better fit can be obtained if anisotropic scatter is assumed. Using the Pitts formula (Equation 9) with both y and x treated as parameters yielded
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c 40
30
RO R
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20
1u
CI
1
4
5
Figure 10. IRozenberg function for methylene blue at 660 nm ( O ) ,610 nm (0),and 390 nm (A)as a function of the reduced concentration, C. The"solidline is a least squares fit to the 610 nm data using Equation
5
the values, 7 6 6 0 = 388, x660 = 1.61, 7610 = 112, and x610 = 2.02. The least squares fit was nonconvergent for the weak band a t 390 nm, showing in fact very little x -dependence. The solid curves of Figure 8 show the optimum fit to the 660- and 610-nm bands.
CONCLUSIONS Because the K-M theory leads to a simple one-parameter equation (Equation 3), it is a very attractive model and is very useful where applicable. In accordance with previous observations (9-5, 9),this study shows that it can be used to quantitatively describe the concentration of an absorbing species in scattering media only over a very limited concentration range. The Rozenberg formula (Equation 5 ) has been derived for strongly absorbing media, and is claimed to give very good agreement with reflectance data obtained using a filter-photometer s,ystem (22). Perhaps the crude filter system masks some effects, for we find that the Rozenberg equation gives a good description of the reflectance data in some cases, but fails completely in others. It appears that the formula is of little value for a weak band, because the slope is too small for Q to be uniquely determined. An examination of Equation 5 shows that, at high concentrations, RoIR should become linear in concentration with slope Qr.On the other hand, it appears that additional difficulties are encountered for strong, overlapping bands as in the case of methylene blue. Because of anomalous dispersion effects, these bands perturb one another and they do not show the expected concentration-dependence of an isolated band. The magnitudes of the Q's which fit our data also raise some questions about the general validity of the Rozenberg theory. According to the approximate parameters of Chekalinskai (Equation 6), we would expect 1 5 Q 5 2, since a r t = 1.
+ +
The value reported by Il'ina and Rozenberg (22) for several systems is 2.3. However, our values of 3.24 and 16.18 are too large to be given any reasonable physical interpretation. The use of an exact solution to the equation of radiative transfer for isotropic scatter offers no advantage over the simple K-M theory. This is not surprising, as it has previously been shown that the solutions differ very little (2). It is significant to observe that solutions which are based upon anisotropic scatter give a much better account of the reflectance of strongly absorbing media. The Pitts formula (Equation 9) has been shown to be a convenient two-parameter equation which fits our data well. It requires large x-values, which implies that the scatter in these model systems is highly anisotropic. In condensed media, the apparent scatter is a result of multiple scattering processes, each of which is known to be anisotropic. It is generally assumed that the net result can be represented by an isotropic phase factor (see, for example, Ref. 17). This is apparently not true of the model systems studied here. It appears that quantitative analysis by diffuse reflectance spectrometry is possible in strongly absorbing media if proper account is taken of the anisotropy of scatter. Some variation in the anisotropy of the bands reported in this study is obvious. There is insufficient data a t present to show whether the degree of anisotropy correlates with the magnitude of the absorbance and/or with the wavelength of the band. Further measurements will be made in an attempt to clarify this point and better define the scattering properties of the model systems. It seems intuitively obvious that the degree of anisotropy within a given elemental sample layer must depend upon the absorbance of that layer. The present reflectance theories do not include a concentration-dependentanisotropy of scatter, but it is suggested that proper account of this effect might lead to an improved model.
LITERATURE CITED (1) H. G. Hecht, J. Res. Nat. Bur. Stand., in press. (2) H. G. Hecht, in "Modern Aspects of Reflectance Spectroscopy", W. W. Wendiandt, Ed., Plenum Press, NBw York, 1968, pp 1-26. (3) W. W. Wendlandt and H. G. Hecht, "Reflectance Spectroscopy", Interscience Publishers, New York, 1966. (4) G. Kortum, "Reflectance Spectroscopy", Springer-Verlag. New York, 1969. (5) R. W. Frei and J. D. MacNeil, "Diffuse Reflectance Spectroscopy in Environmental Problem-Solving", CRC Press, Cleveland, Ohio, 1973. (6) P. Kubelka and F. Munk, Z.Tech. Phys., 12, 593 (1931). (7) A. Schuster, Astrophys. J., 21, 1 (1905). (8) Ref. 4, p 181. (9) G. Kortum and G. Schreyer, 2.Naturforsch.A, 11, 1018 (1956). (IO) E. L. Simmons, Appl. Opt., 14, 1380(1975). (1 1) A. L. Companion, in "Developments in Applied Spectroscopy", Vol. 4, E. N. Davis, Ed., Plenum Press, New York, 1965, pp 221-234. (12) S. Chandrasekhar, "Radiative Transfer", Clarendon Press, 1950 (reprinted by Dover Publications, Inc., 1960). (13) R. G. Giovanelli, Opt. Acta, 2, 153 (1955). (14) E. Pitts, Proc. Phys. Soc., London, Sect. 6,67, 105 (1954). (15) G. V. Rozenberg, Dokl. Akad. Nauk SSSR, 98, 201 (1954). (16) C. A. Lermond and L. 6.Rogers, Anal. Chem., 27, 340 (1955). (17) G. V. Rozenberg, Usp. Flz. Nauk, 69, 666 (1959). (18) E. S. Kuznetsov, I n . Akad. Nauk SSSR, Ser. Geogr. Geofiz., 5 , 247 (1943). (19) V. A. Ambartsumian, Dokl. Akad. NaukArm. SSR,8, 101 (1948); J. Phys. USSR, 8, 65 (1944). (20) I. I. Chekalinskaia, lzv. Akad. Nauk SSSR, Ser. flz., 21, 1494 (1957). (21) H. G. Hecht, Appl. Spectrosc., in press. (22) A. A. Il'ina and G. V. Rozenberg, Dokl. Akad. Nauk SSSR, 98, 365 (1954).
RECEIVEDfor review March 10, 1976. Accepted July 6, 1976.
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