Quantitative Diffuse Reflectance and Transmittance Infrared Spectroscopy of Nondiluted Powders
F A R N A Z B O R O U M A N D , * J A C Q U E S E . M O S E R , a n d H U B E R T v a n d e n B E R G H Laboratoire d'Etude de la Pollution Atmosph~rique et des Sols (F. B., H. v. d. B.), and Institut de Chimie Physique (J. E. M.), Ecole Polytechnique Fgdgrale de Lausanne, CH-1015 Lausanne, Switzerland
An adaptation of Kubelka's general model of diffuse reflectance and diffuse transmittance of light to nonideal scattering samples is proposed. It is applied to quantitative Fourier transform infrared spectrometry of nondiluted surface-derivatized silica powders. Corrections of the mea- surements are introduced in order to take into account common problems that often prevent a truly quantitative application of the DRIFTS ana- lytical method. Effects on the measured data due to the background absorption of the sample and to the specular reflection on its surface are discussed and quantitatively corrected for. The possible existence of gradients of the optical properties of the powder medium is taken into account by a numerical adaptation of the model to inhomogeneities of the phenomenological absorption and scattering coefficients K and S. Model systems constituted of two types of silica powders of very different morphology are prepared. Known concentrations of molecules carrying a cyano functional group as chromophore were covalently anchored on the powder surface. Application of the adapted model gives a quite good description of the measured diffuse reflectance and diffuse transmittance in these cases where the simple Kubelka-Munk function fails.
Infrared spectroscopy is a choice method for the chem- ical analysis of powders and rough surfaces. Many of the solid industrial products of our society, among which are pharmaceuticals, agro-chemicals, plastics, and paper, as well as intermediates encountered in their preparation, appear in an opaque form. Characterization of such com- pounds by a rapid, nondestructive technique may be crucial, for instance, for quasi-continuous quality anal- ysis in a production line. Furthermore, in the analytical laboratory, such opaque powders and rough surfaces are met in many fields, such as chromatography, heteroge- neous catalysis, ceramic technology, and electrochemis- try.
Diffuse reflectance infrared Fourier transform spec- troscopy (DRIFTS) has been established as a powerful tool for the qualitative analysis of such samples. In par- ticular, the pioneering study by Griffiths and co-workers has demonstrated the high potential of DRIFTS for qual- itative analysis in a wide range of applications. 1-7 The possibility of using the same technique for quantitative analysis was also recognized. ~,3,4,s,9 Whereas applying DRIFTS to qualitative chemical analysis of powders is relatively simple, many theoretical and practical prob- lems appear when one is attempting quantitative anal- ysis of functional groups.
Received 1 May 1992; revision received 7 July 1992. * Author to whom correspondence should be sent.
The effects of several experimental parameters such as the particle size) ,1° the packing density, and the pres- sure applied to the surface of the powder in preparing the samples) 1-~3 as well as the angles of incidence and collection of light on the diffuse reflection intensity of loosely packed powders, have been studied. 14,15 The im- portance of taking into account all these parameters to achieve quantitative measurement has been pointed out in the literature.
Conventionally, a simplified equation, derived from the Schuster 16 and Kubelka-Munk 17,1s phenomenological theory of light absorption and scattering of powder layers in which the sample is treated as a continuum, is used in order to relate a chromophore concentration to the intensity of the sample diffuse reflection:
• (R~) = (1 - R ~ ) 2 / 2 R ~ = K / S ~ c. (1)
In Eq. 1, R~ represents the ratio of the diffuse reflectance of the sample to that of a selected reference. K designates an absorption coefficient proportional to the concentra- tion c of the chromophore, and S the scattering coeffi- cient of the powder. 1°
In such a model, the first assumption made is that the sample thickness d is large enough (d ~ ~ ) to allow the contribution of light transmission through the layer and the reflection from the background situated behind it to be neglected, i.e., R -~ R~. The second assumption re- quires that both phenomenological parameters K and S be taken to be constant throughout the whole sample volume. In general, these two important assumptions are not valid for real systems. A peculiar approach has there- fore prevailed so far, in which the sampling method is actually adapted to satisfy the conditions of this deficient theoretical model. This strategy is done in the following way: a nonabsorbing and highly scattering matrix of an alkali halide, such as KBr or KCI, is used in order to dilute the powder that is to be analyzed as well as to minimize spectral distortions. The same diluent is then used as pure powder as the reflectance reference. The distortions in the spectra observed in highly absorbing samples arise because of an increase in the fraction of the radiation undergoing Fresnel-type reflection from the front surface of the sample as compared to the frac- tion that has been refracted and scattered through the sample and contains absorption information related to the concentration of the species to be measured? 3 The dilution in alkali halides is also used to satisfy the con- ditions for the use of Eq. 1, i.e., the achievement of an optically thick layer within a reasonable depth of, say, a few millimeters, and that of macroscopic homogeneity of the sample, ensuring that the phenomenological ab-
sorption and scattering coefficients are constant through- out the sample.
The preparation of reproducible and homogeneous mixtures of the powder to be analyzed with the diluent is a critical operation for quantitative work. The effects of the sample preparation procedure on the homogeneity of the mixture, and hence on the intensity of the ab- sorption bands, have been measured, and the respective merits of several powder sample preparation methods have been discussed. 9 In many cases both the difference in grain size, between the powder to be analyzed and the diluent, and the possibility of selective aggregation of sample particles, due to the hygroscopic properties of the halide and electrostatic interactions between insulating particles, render the preparation of close to ideal mixtures extremely difficult. Moreover, the dilution may decrease the detection sensitivity in systems where the chromo- phore is highly dispersed and its optical absorption is low. Of course one would also prefer to avoid diluting powders, as it is evidently desirable to analyze samples that are as close as possible to their original state.
Our own effort in this domain has focused on the quan- titative spectroscopy of nondiluted powders. To attain this goal, we have chosen as model systems inorganic powders--such as, for instance, silica gels--in which a known quantity of a chosen chromophore has been at- tached covalently to the surface. Furthermore, samples of finite thickness are considered. The possibility of the existence of profiles of both the absorption and scattering coefficients within the depth of the layer is introduced. Also, the problems due to the background absorption by the substrate material and the Fresnel-type reflection are taken into account. Several correction and calcula- tion methods are proposed, which are required to adapt a model, based on the general hyperbolic solutions of the Schuster and Kubelka-Munk set of differential equa- tions, to the quantitative analysis of such nonideal sys- tems.
Commercial silica powders with various morphologies are available which range from fluffy aerogels to densely packed powders. Chemical surface derivatization of such materials plays an important role in the preparation of stationary phases used in gas and liquid chromatogra- phy. 19 The quantitative determination of small surface concentrations of organic molecules covalently linked to the oxide surface is not trivial. 2° For the application of diffuse reflectance infrared spectroscopy, very loose de- rivatized silica powder layers are hardly optically thick enough. Their high compressibility and their tendency to aggregate when mixed with an alkali halide diluent, because of electrostatic and hydrophobic interactions, render their quantitative analysis very hazardous. In- deed the numerous problems arising in the application of DRIFTS to such materials certainly qualify surface- derivatized silica powders as choice model systems to study for sample nonideality.
In our approach, standard samples consisting of two types of chemically pure silicon oxide (Cab-O-Sil and LiChrosorb) which have quite different morphologies are prepared. The surface hydroxyl groups of the silica sub- strate are derivatized by a silylation reaction with an aminosilane that carries a suitable chromophore. Thus, known concentrations of molecules containing a cyano
functional group are anchored on the powder surface. Concentrations of surface chromophore are then trans- lated into absorption coefficients that are introduced into the adapted model in which the main nonidealities are taken into account.
Diffuse transmittance is measured for optically thin samples with the use of a novel photopyroelectric detec- tor. This device enables a dual-channel sequential mea- surement on the same sample. Hence, the number of the observables is increased, whereas the number of the pa- rameters to be adjusted remains the same. The calculated diffuse reflectance and transmittance are finally com- pared to experimental values.
ADAPTATION OF KUBELKA'S THEORETICAL MODEL OF DIFFUSE REFLECTANCE AND TRANSMITTANCE TO THE POWDERS' NONIDEALITY
Four important parameters that can be considered as being characteristic of the nonideality of a sample sub- mitted to DRIFTS analysis are discussed here.
(1) Finite Sample Thickness. In a nondiluted sample, the semi-infinite depth condition is not necessarily achieved within a typical conventional sample depth of 2 to 3 ram. The minimal depth beyond which there is essentially no transmittance of the light through the sam- ple (i.e., doo is attained), required for application of the simplified Eq. 1, depends on the absorption and scat- tering coefficients of the system, and it should be deter- mined in each particular case. If the K and S parameters are relatively small, do~ may exceed 5 millimeters. It is therefore reasonable in a more general case to allow for partial transmission of light through a sample of finite thickness d. Kubelka's hyperbolic Eqs. 2 and 3 then pro- vide the necessary theoretical model relating diffuse re- flectance and transmittance to both phenomenological parameters K and S of the sample: 1°,18
R = 1 - Rg (a - b . c o t h ( b S d ) ) (2) a + b . c o t h ( b S d ) - R e
b T = (3)
a . s i n h ( b S d ) + b . c o s h ( b S d )
where R is the diffuse reflectance, T is the diffuse trans- mittance, a -= 1 + K / S , b =- (a 2 - 1) 1/2, and Rg represents the proportion of transmitted light that is returned into the sample by partial reflection of the background if one is present behind the sample layer.
(2) Background Absorption. In the general case, the substance to be analyzed in a sample is not present in a pure form. Some other compounds, the spectra of which may overlap with that of the chromophore of interest, can therefore also absorb at the wavelength chosen to perform the quantitative measurements. If the concen- tration of the species to be analyzed is the only one that varies, other species would yield a constant background absorption. Since Eqs. 2 and 3 provide only absolute values of R and T, the problem cannot be simplified by using a sample of zero concentration of the chromophore as a reference and by measuring reflectance and trans- mittance relative to it. The background absorption has
APPLIED SPECTROSCOPY 1875
therefore to be taken into account and a total absorption coefficient defined by Eq. 4:
K ~ = K + Kbackgroun d. (4)
The phenomenological absorption coefficient K (cm -1) then can be simply related to the mass concentration cm (mol. g-~) of the analyte molecules in a sample of density p (g. L -1) and to their Beer-Lambert decadic molar ex- tinction coefficient ~ (L.mol 1.cm-~) by Eq. 5.1°
K = 2"ln(lO)'e'p'Cm. (5)
(3) Fresnel-Type Reflection. Light impinging on the surface of a powder is reflected in various ways. The measured total reflectance therefore consists of several fractions that are distinguished by the physical phenom- enon from which they originate and by the laws governing their relative intensity. The easiest fraction to be con- ceived intuitively is the one given by the external reflec- tion of light on the macroscopically flat surface of the sample. This portion is generally called specular or Fres- nel regular reflection. It is characterized by equal angles of incidence and reflection. In a powder, however, indi- vidual faces of the particles placed in the "top layer" are not all aligned with the sample's macroscopic surface. Because of the arbitrary distribution of their orienta- tions, reflection by these faces is not necessarily con- tained in the incidence plane and appears as partially diffuse. Since in this case the reflected light has not pen- etrated the particle, its intensity still obeys Fresnel's law. The term dif fuse reflection is normally used to define the remission of that fraction of the incident radiation which has penetrated and has been reflected by at least one particle. The direction of this kind of reflection is no longer predictable just from the surface geometry and the incidence angle. This second contribution to the re- flectance of a sample (R~) is the only one the Kubelka- Munk theory is treating.
Let us consider at this point diffuse radiation imping- ing on a powder or any rough surface. A part R~ of the original incident intensity is reflected, according to the Fresnel law, on the macroscopic surface of the sample as well as on the individual tilted faces of directly exposed particles. As a consequence, only a fraction (1 - RF) of the incident intensity does in fact penetrate the layer and is submitted to diffuse reflection. The measured overall reflectance can be written therefore as the sum of both portions of radiation RF and R~. (1 - RF) that follow the Fresnel law and Eq. 2, respectively:
R . . . . : R,-(1 - RF) + RF. (6)
The Fresnel regular and Fresnel diffuse reflection con- tributions can be removed to a large extent by the use of an off-axis collection geometry and/or crossed polar- izers, and by a specific blocking plate included in some diffuse reflectance accessory designs. In this case, al- though the last term of Relation 6 is eliminated, mea- sured reflectance still depends on Fresnel reflectivity of the sample.
Since absolute values of the reflectance have to be fed into Eq. 2, diffusely reflected intensities should be nor- malized by that of an optically thick nonabsorbing stan- dard layer, the reflectance R~ '~f. of which could be as- sumed as equal to unity. The external reflectance of a
nondiluted sample is not necessarily the same as that of such a reference layer. RF values have therefore to be determined in both cases in order to relate the measured diffuse reflectance R . . . . to the purely diffuse reflectance R ==- R~ treated by Kubelka's model equation:
Diffuse transmittance measurements should, in a sim- ilar way, take into account the attenuation of the incident light by a factor of (1 - RF). Real transmittance of the sample layer, described by Eq. 3, could therefore be ob- tained after suitable correction of the measured value:
T = Tme~." (1 - R~) -1. (8)
A rough measurement of the Fresnel reflectance of a sample can be performed with any conventional optical collection accessory. If one assumes that the linear po- larization of the incident light is conserved when it un- dergoes specular reflection on a surface and, on the con- t rary, is tota l ly scrambled by mult iple reflection, refraction and diffraction phenomena occurring inside the scattering-layer, crossed polarizers can be used in order to cut the external reflection contribution off. In this case, the light intensity measured at the exit of the accessory, right after the second polarizer, can be written as :
I . = Ii '~'R~' (1 - RF) (9)
where Ii represents the intensity of polarized light im- pinging on the sample surface, and ~ is a factor including the reflected radiation collection efficiency as well as the second polarizer's transmission of randomly polarized light. Turning the first polarizer, placed in front of the sample by 90 ° , in order to align its plane of polarization with that of the second polarizer, causes the contribution RF of the Fresnel reflection to be added to the total measured reflected intensity.
I,, = Ii '~'[Rd'(1 - RF) + R~]. (10)
When such an experiment is run with the standard layer of unit reflectance (RJ ee- = 1), Eq. 10 immediately yields the equality Ii '7 = I,,ref" The expression of the Fresnel reflectivity of the sample can therefore be written as :
R~ = (Iu - I.)/IIL "el" (11)
(4) Nonhomogeneity of Absorption and Scattering Co- efficients. Among the set of hypotheses made in order to derive Kubelka's equations, it is assumed that absorption and scattering coefficients are constant throughout the whole sample layer. In a powder, which is essentially a discontinuous medium, these conditions imply homo- geneous absorption and scattering properties on a mac- roscopic level and therefore the constancy of the con- centration of the chromophore and of the number of air/ solid interfaces per volume unit. In reality, several pa- rameters can affect the homogeneity of optical coeffi- cients (K, S) in the sample. The importance of geometric irregularities at the surface and inside the layer is dis- cussed elsewhere. 21
Some other phenomena can cause a continuous vari- ation of the optical properties of the powder from the
1876 Volume 46, Number 12, 1992
top surface to the bottom interface. These can be de- scribed by profiles of one or both of the phenomenological parameters K and S varying with the depth of the layer. Most powder samples are characterized by a distribution of particle size. In a fluidized medium, segregation of grains with regard to their respective diameters is likely to occur along the vertical axis. Since the specific surface area of smaller particles is larger, this segregation can lead, for example, to a macroscopic gradient of the con- centration of molecules that are adsorbed at a given sur- face concentration on the solid surface and, consequent- ly, to a profile of the effective absorption coefficient. Although the variation of the scattering coefficient is more difficult to predict quantitatively, it should also be affected by the effect described above, which modifies the distribution of interfaces within the depth of the layer.
Applying a pressure to the surface of a loose powder sample results in a more or less dense packing. Attenu- ation of the stress in the medium yields a three-dimen- sional distribution of the apparent density of the ma- terial. Several parameters are important for the definition of the final state of such a powder submitted to a given pressureY 2"~ The pressure applied and the geometry of the sample holder, as well as the shape of the particles and the internal friction, can be quoted as examples. Many theories have been established in order to try to relate all these parameters to the packing of the dispersed mediumY 2e The existence of a multitude of such attempts at modeling this phenomenon indicates the complexity of the problem. Applying a pressure to the horizontal surface of a powder sample to obtain its macroscopic flattening is a very common procedure. Contrary to the technique employing a spatula to scrape the sample sur- face, the use of a press introduces only vertical con- straints in the powder. Although a radial gradient of the density inevitably appears as a consequence of the fric- tion on the walls of the container, a one-dimensional compression profile may be assumed at the center of the sample, where the analyzing incident beam of the spec- trometer is focused.
Under these conditions, the problem caused by the lack of homogeneity of the optical properties of the me- dium can be reduced to the integration of the Schuster and Kubelka-Munk set of differential equations con- taining K and S coefficients which are functions of the vertical coordinate. Since there is no simple analytical solution for the general case, 23,24 modelization of diffuse reflectance and transmittance then requires a numerical resolution of the set of equations.
The algorithm used in this work is easier to implement (yet as rigorous) than classical numerical integration pro- cedures. It has proven in practice to yield results of com- parable precision within shorter computation timeY 4 In- stead of one's trying to directly solve the set of differential equations, the sample is conceptually divided from the top surface to the bottom into a series of horizontal in- finitesimal sublayers of thickness Az. Absorption and scattering coefficients vary from one stratum to the other in accordance with the analytical expression of their profile in the depth of the powder. Az is taken to be sufficiently small, so that within each quasi-differential sub-layer K and S can be assumed to be constant and
Kubelka's hyperbolic solutions can be applied, providing d = Az and, except for the very last layer, Rg = 0:
R = s inh(b .S .d) (12) a . s inh(b .S .d ) + b .cosh(b .S .d)
b T = (13)
a . s inh(b .S .d ) + b .cosh(b .S .d)
where a and b have the same meaning as in Eqs. 2 and 3. The problem is then reduced to the evaluation of the
total diffuse reflectance and transmittance of a stack of n parallel layers, for each of which individual R and T values are known. For two sublayers, a solution has al- ready been reported: 25
T,,2 = T~T2(1 + RIR2 + R~R~ + . . . )
T~T2 - ( 1 4 )
1 - R1R2
2 2 R1.2 RI + T~R2(1 + R,R2 + R1R2 + . . . )
T[R2 = R, + 1 - R~R2" (15)
This solution can be iteratively generalized to the case of a system constituted by j sublayers. The first j - 1 upper layers are likened to a unique one, characterized by the quantities R u _ 1) and Tu _ 1). The process of taking into account an additional layer j can be achieved in a similar fashion by using Eqs. 16 and 17 for the global reflectance and transmittance of the stack:
R(j) = R(j_I) + T=v_,~'R~ (16) 1 - R( j_ , .R i
T(j_I). Tj T(i~ - 1 --- ~ :R j" (17)
INSTRUMENTATION AND MATERIALS
A Bomem DA3 Fourier transform infrared (FT-IR) spectrometer was employed to obtain data throughout this work, in conjunction with a short 25-cm scan tube, specially designed for the low mirror velocities inherent in the use of the slow pyroelectric detector that is em- ployed in the diffuse transmittance measurements. A photopyroelectric detector assembly was used as the sample holder for diffuse reflectance (DR) and as detec- tor for the novel diffuse transmittance infrared Fourier transform spectroscopy (DTIFTS). The principle and details of the photopyroelectric detector have been re- ported elsewhereY 6-2s The sample holder/detector used in this work consists of a thin (28-#m) Kynar polyvi- nylidene fuoride (PVDF) film (Pennwalt Corp., Valley Forge, PA) sputter-coated on both sides with ~ 250 nm of Ni-A1 layers, and inserted in an Inficon microbalance housingY s The PVDF photopyroelectric element used for these experiments has an active area of circular cross section and a diameter of 9 mm. In order to eliminate contributions from light backreflected directly by the metallized PVDF surface into the powder, great care was taken to blacken that surface by depositing a thick layer of carbon soot on it (~ 100 #m). The sooted surface was
APPLIED SPECTROSCOPY 1877
checked to generate zero-level signals in the DA3 InSb diffuse reflectance detector. The sample holder was me- chanically adapted and fitted to the "Praying Mantis" diffuse reflectance accessory (Harrick Scientific Corp., Ossining, NY, Model IMG 2700 L). Two wire grid KRS-5 polarizers (Harrick Scientific Corp.) were mounted in the analyzing beam of the spectrometer, one just in front of the planar entrance mirror of the accessory and the other after the exit mirror. An Ithaco Model 1201 (Ithaco Corp., Ithaca, NY) low-noise, wide-bandwidth preamplifier was used to amplify the photopyroelectric signal to a level compatible with the A/D converter of the DA3 instru- ment (~0.1-10 V). Since the photopyroelectric detector response increases drastically at low frequencies, a small variation in the interferometer scan speed would result in intolerable fluctuations of the detector signal. A high- pass first-order electronic filter is therefore added to equalize the detector response at low modulation fre- quencies and so improve the flatness of spectral base- lines. A second amplifier compensates for the electric signal attenuation due to the filter by finally delivering to the A/D converter a corrected signal, the amplitude of which does not vary with modulation frequency by more than 30 % between 10 Hz and 1 kHz. DRIFT and DTIFT spectra were obtained sequentially by switching connections from the InSb detector and photopyroelec- tric circuit to the input board of the spectrometer, with- out disturbing the powder between the two measure- ments. Diffuse reflectance spectra were normalized by the 100% reflecting, semi-infinite-thickness, dry KBr layer spectrum. Diffuse transmittance spectra were nor- malized by the blackbody spectrum of the empty, soot- coated PVDF sample holder/detector.
Two kinds of silica powder were employed in this work. The precipitated silica LiChrosorb Si 100 (Merck, Darm- stadt, Germany), characterized by a particle diameter ranging from 5 to 10 #m, was dried at 120°C under vac- uum prior to being used. The fume silica Cab-O-Sil M5 (Fluka, Buchs, Switzerland) is distinguished by a pri- mary grain diameter of 10-15/~ and a secondary particle size of the agglomerated small particles on the order of 100/~.19a This nonporous powder is submitted to con- secutive thermal and hydrothermal treatments in order to obtain a reproducible surface for chemisorption. 2' Both silicas were then stored and manipulated in a glove-box under a dry argon atmosphere, at a humidity level not exceeding 1 ppm. The specific surface area (SA) of both silica powders was determined by evaluation of the N2 adsorption BET isotherm at 77 K. Three independent measurements were performed with a Sorptomatic 1900 instrument (Carlo Erba, Milan, Italy) in a relative pres- sure domain ranging between 0.05 and 0.23. Average re- sults of SA = 321 + 3 m2"g -~ and SA = 191 _+ 2 m2.g -1 were obtained, respectively, for the porous LiChrosorb silica and the nonporous Cab-O-Sil.
Hydroxyl groups on the oxide surface can undergo silylation reactions with dimethyl-aminosilanes. Li- Chrosorb and Cab-O-Sil powders were chemically treat- ed with two different reagent compounds. 2°b,2~ (5-Cyano- 3,3-dimethyl-pentyl)-(dimethylamino)-dimethylsilane (DMP.CN) contains a cyano functional group with a rel- atively sharp infrared absorption band peaking at ~ 2247 cm -~ that is quite characteristic. (3,3-Dimethylbutyl)-
(dimethylamino)-dimethylsilane (DMB) has approxi- mately the same structure as DMP.CN, but does not carry a cyano chromophore. KBr (Merck, puriss, p.a. grade) powder was finely ground in a Wig-L-Bug mill capsule (Crescent Manufacturing Co., Chicago, IL) down to a diameter of 20 ttm or less and dried for several hours in an oven at 180°C just before being used as a reflectance standard.
PRELIMINARY EXPERIMENTS AND CHARACTERIZATION OF MODEL SYSTEMS
Testing of the Rejection of Fresnel-Type Reflection by Crossed Polarizers. Although the "Praying Mantis" op- tical accessory does not collect all the diffusely reflected radiation and ideally only an integrating sphere will allow the true diffuse reflectance (DR) spectrum to be mea- sured, the DR spectra collected between 0 ° and 90 ° with our instrument are expected to be representative of the true diffuse reflectance within reasonable error due to demonstrated angular dependence of DR spectraJ 4,15,29 This expectation was substantiated by the very good agreement between DRIFT and DTIFT spectra of thin Probimide 408 films, 28a where the latter is not subject to the instrumental limitations of the accessory.
Brimmer and Griffiths have questioned the use of crossed polarizers to eliminate the specular component of reflectance spectra. 14,15 These authors concluded that, with the off-axis accessory they used, crossed polarizers tended to increase the fraction of specularly reflected light collected and detected. A simple preliminary ex- periment has been performed in order to test the char- acteristics of our "Praying Mantis" accessory, used in conjunction with polarizers, as well as to check the va- lidity of the approach described above for Fresnel-type reflection measurement and correction. In an effort to simplify the procedure and secure a perfect alignment, measurements have been performed on an optical bench outside the FT-IR spectrometer with the use of visible radiation. Indeed, the behavior of the accessory with re- spect to light polarization should not depend on the wavelength. A xenon arc lamp source and a photomul- tiplier tube detector were employed. The collimated beam of white light was polarized by being passed through a Polaroid film and then focused on the carefully aligned "Praying Mantis" optics. The intensity of the radiation reflected by the sample surface and collected by the el- lipsoidal mirror of the accessory was measured after the beam passed through a second polarizer placed in front of the detector. Results obtained for various polarizer configurations are reported in Table I for two different samples. One of them was constituted of a coarsely pol- ished aluminum strip, while the other was made out of a plate of rough colorless PTFE.
In a first measurement, achieved without putting any polarizer in the light beam, it was established that the global reflectance of the metallic shining strip is about 60 % higher than that of the rough polymer surface. In- sertion of a polarizing filter in front of the entrance mir- ror of the accessory caused the global measured trans- mittance of the device to diminish to a quarter of its initial value. This attenuation, that has approximately
1878 Volume 46, Number 12, 1992
I 0 I
TABLE I. Behavior of the "Praying Mantis" optical accessory towards light polarization. Collected light intensities I for shiny aluminum and rough PTFE samples are given in arbitrary units. Directions of planes of polarization of polarizers P,, P2 and P3, represented in the scheme above, are indicated by vertical (]) and horizontal (~ ) arrows. Symbol 0 means the absence of a polarizer.
I I/I o P1 P2 P3 I (A1) (PTFE) I/Io (A1) (PTFE) 0 0 0 1620 1000 1 1 T 0 0 409 264 0.25 0.26
0 0 421 251 0.26 0.25 ~ 0 220 136 0.14 0.14 0 ~ 187 79 0.12 0.08 0 T 42 56 0.03 0.06 1 o o o o o
the same magnitude for both samples, was not affected by rotating the polarization plane imposed to incident light and can therefore be ascribed to the sole polarizer. The transmission factor T~ of this filter can be artificially expressed as the product T1 = x I • T H of the fraction x H of the intensity of incident light characterized by a polar- ization plane parallel to that of the filter and the trans- mittance T H of the filter for this fraction of the radiation. The addition of a second polarizer in front of the acces- sory, set in the same direction as the first one, and the measurement of the global resulting transmittance T2 = T~.Tll, afforded a handle for direct evaluation of these two parameters that were found to be xjl ~ 0.5 and T H = 0.54. Moving one of the polarizers in front of the detector induced a different behavior for each of the two reflective surfaces. Although in the case of the aluminum strip the new measured transmittance T3 is only slightly inferior to T2, only 57% of the previously collected intensity could be obtained with the PTFE plate. This observation therefore indicates that the fractions x = (1 - T3/T2)/ xl, ~ 0.3 and 0.9 of the incident light were depolarized by passage through the optical accessory and reflection on each of the respective surfaces. Rotation by an angle of 90 ° of the second polarizer caused a much more pro- nounced attenuation for the metal surface, on which Fresnel reflection tends to retain the light polarization, than for the polymer sample. In both cases, transmit- tance T4, measured in this configuration, verifies ap- proximately the relation T2 = T3 + T4.
A comparison of results obtained for the shiny metal surface, where the specular component of reflection should dominate, and for the rough PTFE plate, where, to the contrary, a predominant ideally diffuse reflection is expected, allows us to draw two conclusions: (1) The polarization of light seems to be retained by passing
through the "Praying Mantis" optical accessory. Rota- tion of 90 ° of the polarization plane by the reflection on the first ellipsoidal mirror is apparently compensated for by further rotation during reflection on the second one. (2) Although most of the regularly reflected light should be eliminated by the right-angle off-axis geometry of the collection optics, a large contribution to the measured intensity due to (diffuse) Fresnel reflection seems to re- main. This harmful contribution to the analyzed radia- tion is indeed efficiently removed in this case by the use of crossed polarizers.
I n d e p e n d e n t D e t e r m i n a t i o n o f A b s o r p t i o n C o e f f i c i e n t s . A controlled variation of the quantity of a chromophore, represented in our model systems by the cyano functional group, is obtained by preparing powder samples con- taining various concentrations of DMP.CN surface sub- stituents. The concentration of siloxy groups covalently linked to the surface of a silica treated by complete re- action with the aminosilane DMP.CN is determined by carbon elementary analysis. Such a characterized powder can then be mixed in known proportions to the powder treated with DMB, which does not contain any cyano chromophore and which will act therefore as a diluent. Similarly to the case of the dilution of a derivatized silica in an alkali halide, mixing a powder covered by substit- uents of low polarity with an untreated oxide, charac- terized by a high-energy surface, will lead to the for- mation of fatal inhomogeneities in the sample to be analyzed. Specific aggregation of one of the powders in the sample can be avoided only if the respective surfaces do not have a sufficient affinity with each other. This problem has been solved in our model systems by prep- aration of mixtures of two silicas of the same type, the surface of which has been treated to the point of complete coverage by reaction with either DMB or DMP.CN ami- nosilanes, respectively. Powders covered by such a hy- drophobic layer, moreover, do not adsorb much humidity and can therefore be more easily manipulated in the lab's atmosphere.
For a chromophore dispersed homogeneously in a non- absorbing matrix, the absorption coefficient K can be expressed as a function of the mass concentration cm by Relation 5. In our model powders, the mass concentration of the substituent carrying the cyano functional group is defined by Eq. 18:
w-F cm = (18 )
1 - - + r - ( M - 5) SA
where w is the weight fraction of the silica carrying -C-=N groups in the powder mixture, M the molecular weight of DMP.CN siloxy groups, and SA (m2"g -1) the specific surface area of the naked silica. The surface con- centration F (mol.m -2) of aminosilane substituents is determined by measuring the nitrogen and carbon con- tent of the powder by micro-elemental analysis (CHN) of typically 10 mg of derivatized silica with the use of a Perkin-Elmer 2400 Elemental Analyzer. ~ ~ 5 g-mo1-1 represents a correction factor, taking into account the molecular weight of the proton and of adsorbed water displaced from the hydroxylated oxide surface by the silylation reaction. 21"
APPLIED SPECTROSCOPY 1879
TABLE If. Macroscopic absorption coefficients determined at ~ at 2247 cm -~ for two silica powders entirely covered by DMP.CN substituents. Explanation of the symbols is given in the text.
Sa F p Silica type (m2/g) (mol/m ~) (g/L) K (cm-') Kb (cm 1)
The analyzed powder densi ty is easily calculated from the sample container volume and the powder mass mea- sured by weighing the sample holder. Typical values ob- ta ined are p = 370 g/L and p = 130 g/L for LiChrosorb and Cab-O-Sil powders, respectively. The molar extinc- t ion coefficient E (mol-~.L.cm -~) of the chromophore is therefore the only unde te rmined parameter still remain- ing in Relat ion 5. Various classical methods have been tenta t ively applied to measure quant i ta t ively the ab- sorpt ion spectra of derivat ized silica in conditions where light scattering could be neglected. Th in s intered disks p repared with KBr as a di luent were never sufficiently t ransparent , homogeneous, or s turdy enough to allow direct- t ransmission IR measurements . The immersion method, where the solid is dispersed in a liquid of match- ing refractive index, did not give the expected results. Instabi l i ty of the powder dispersions in thin liquid films required by infrared spec t romet ry lead to unacceptable uncer ta int ies in the de te rmina t ion of the chromophore concentra t ion as well as of the optical pathlength. Large variations in the intensi ty of absorpt ion bands of various cyano-alkanes have been repor ted as being due to solvent and subs t i tuent effects) ° The average oscillator s t rength of the C=-N group depends primari ly on its electronic polarization: the higher the ( ~ + ) C - N ( ~ - ) dipole mo- ment , the larger will be the effective cross section for infrared absorption.
The molar ext inct ion coefficient of the - C - N func- t ional group carried by a molecule in solution at the characterist ic wavelength of its absorpt ion in the mid- infrared is affected by several factors related to its en- vi ronment . I t has been observed generally tha t hydrogen bonds formed with the cyano group, as well as inductive effects of e lect ron-at t ract ing subst i tuents , can substan- tially increase the absorpt ion band intensity. 3~ On the derivat ized surface of our silicas, the cyano group envi- ronmen t is principally const i tu ted of alkyl chains of neighboring subst i tuents and of their nitrile end groups. The very close proximi ty of molecules anchored on the solid surface must, moreover, correspond to densities similar to those in the condensed phase of a pure liquid. Inf rared transmission spectra of 50-#m-thick liquid films of pure cyano-heptane (CsH~N), cyano-octane (CgH~TN), and caprilonitri le (C~oH]9N) allowed the de terminat ion of their molar ext inct ion coefficient at p = 2247 cm -] as, respectively, ~ = 22, 23, and 27 mol -~. L. cm-L In the same exper imenta l conditions, ~ = 34 mol-~-L.cm -] was ob- ta ined for the pure D M P . C N aminosflane compound. This higher value may be ascribed to hydrogen bonding between the amino groups and the electron-deficient car- bon of the -C=-N moiety. Obviously, it cannot be used for the siloxy radical subst i tuents (C10H2oNSi) l inked to the oxide surface, which have lost their amine group during silylation reaction. Very close s t ructural similar-
z o_ CD
o b- CO
O O
O b- O - r D.
b)
3200 2800 2400 2000 1600
v [ cm-1]
FIG. 1. FT-IR photoacoustic spectra, recorded in an MTEC Model 100 cell filled with air at atmospheric pressure, for various powder samples: (a) dry standard KBr reference powder; (b) LiChrosorb silica treated up to its surface saturation with the aminosilane DMP.CN; (c) Cab-O-Sil silica covered in the same way by nitrile substituents. The abscissa represents in each case the true zero level of the photoacoustic signal. Dashed lines, representing the interpolated signal levels due to the oxide background absorption alone, are drawn between experi- mental points obtained at ~ = 2200 and 2400 cm-'.
i ty existing between these subst i tuents and caprilonitri le suggests t ha t their molar ext inct ion coefficients should be about identical, e = 27 mo1-1. L -cm -1 will therefore be used for our chromophore at the wavelength of the max- imum of its characterist ic absorpt ion band at ~ = 2247 cm -1. Typica l absorpt ion coefficients for cyano-siloxy groups anchored on surface derivat ized silica powders de te rmined in this way are tabula ted in Table II.
A b s o r p t i o n D u e to t h e S i l i c a S u b s t r a t e . Pre l iminary exper iments conducted with mixtures of SiO2 powders derivat ized by t r e a tm en t with DMB and DMP.C N showed tha t a residual absorpt ion remains as the chrom- ophore 's concentra t ion goes to zero. Absorpt ion of the silica substrate itself has therefore to be taken into ac- count for the de te rmina t ion of total absorpt ion coeffi- cient K. Photoacoust ic spectroscopy yields signals t ha t are propor t ional to the t rue absorpt ion coefficient of a powdered sample, provided tha t this absorpt ion is not too high and no sa tura t ion occurs. Quant i ta t ive mea- surements can therefore be per formed by using an in- ternal reference, 13 thus allowing for an independen t de- te rmina t ion of the silica suppor t absorption. Since in our case bo th the concentra t ion and the absorpt ion coeffi- cient of the - C - N chromophores have been established, these can be used as a reference. Figure 1 shows F T - I R photoacoust ic spectra obta ined for samples placed in an M T E C 100 closed cell. Ro-vibrat ional bands centered at
= 1600 cm -1 and p = 2350 cm -1 can be observed, which are due to water vapor and CO2 present in the atmo- spheric air contained in the cell. Spec t rum a has been recorded for the finely ground K Br powder employed thereaf te r as a 100% reflective s tandard for diffuse re- flectance normalization. No significant absorpt ion at all could be measured in this case between ~ = 1200 and 3200 cm-L Such an observat ion substant ia tes the choice of this powder as a nonabsorbing reference and d e m -
1880 Volume 46, Number 12, 1992
W {J Z < I'..- o U.I ,_1 I J_
n -
_B /
_A
4000 3500 3000 2500 2000
~" [ cm -1]
FIG. 2. Raw diffuse reflectance spectra of 3-mm layers of LiChrosorb powders entirely covered by DMB (A) and DMP.CN (B) siloxy sub- stituents. Number of coadded scans: 250; resolution: 4 cm-~; interferom- eter scanning speed: 0.5 cm/s.
onstrates the quality of the baseline offered by the photo- acoustic technique. Spectra b and c were obtained, re- spectively, for LiChrosorb and Cab-O-Sil powders, treated up to their surfaces saturation with the DMP.CN amino- silane. Both absorption bands centered at p = 2247 cm -1 and around 3000 cm -1 are characteristic features of the cyano-siloxy group. Silica background absorption could be extracted in these two cases from the base level of the absorption peaks. The origin of light absorption by the silicon dioxide material itself has not been yet clearly identified. However, one may assume that the wide ab- sorption bands appearing below 2100 cm -1 and above 2700 cm -1, which are probably due to underivatized hy- droxyl groups and water traces absorbed on the silica surface, could extend to, and cause significant absorption in, the region around 2250 cm -1. The absorption coeffi- cient corresponding to the height of the peaks centered at P = 2247 cm -1 has been determined from Eqs. 5 and 18 and by taking into account ~ = 27 + 1 mol-l .L.cm -1 as K = 44.5 +_ 2 cm -1 for the - C - N chromophore dis- persed on LiChrosorb and K = 12.2 + 0.4 cm -~ on Cab- O-Sil powders (Table II). Absorption coefficients of these two silica substrates are evaluated from these values as, respectively, Kb = 13 _+ 0.5 cm -~ and Kb = 3 + 0.2 cm -1. If we take into account the difference between the ap- parent density of both powders (0.35 and 0.12) as well as their different specific surface areas ( S A = 321 and 191 m2/g), absorption of the dense porous LiChrosorb ap- pears nearly exactly equal to that of the loosely packed, nonporous Cab-O-Sil. This result further substantiates the assumption that the absorption of surface adsorbed hydroxyl groups and water traces are at the origin of the powder substrate background absorption.
EXPERIMENTAL RESULTS AND DISCUSSION
Figure 2 shows raw diffuse reflectance spectra of two LiChrosorb powders, with their surfaces saturated by reaction with the DMB and DMP.CN aminosilanes, re- spectively. Homogeneous mixtures in various measured fractions of these two silica powders are prepared by shaking both weighed fractions together in the Wig-L- Bug capsule deprived of its grinding ball. A weighed amount of the resulting powder mixture is then poured
0.15 i i I , l I i i I i i I 2300 2270 2240 2210 2180
V [ cm -1]
FIG. 3. Relative DRIFT partial spectra of 3-mm-thick samples of mixtures of two LiChrosorb silica powders entirely covered by DMB and DMP.CN substituents, respectively. The wavenumber domain is restricted to the characteristic cyano group absorption peak. Reflec- tance is normalized to tha t of a standard 3-mm dry KBr layer. Labels represent for each spectrum the corresponding weight fraction w of DMP.CN-derivatized powder in the mixture. Spectrometer operating conditions are the. same as in Fig. 2.
through a funnel into the 3-mm-deep container of the sample holder. The surface of the powder is finally lev- eled with the top of the cup by letting a stainless steel piston driven by a dynamometric press vertically down on the sample. Raw diffuse reflectance values are deter- mined at the - C - N group maximum absorption wave- number at ~ = 2247 cm -1 for eighteen different powder mixtures, for which the weight fraction w of DMP.CN- derivatized LiChrosorb is varied from zero to unity. Since no light transmission through the powder samples can be measured by the photopyroelectric detector, the semi- infinite thickness condition in this case is truly verified. Figure 3 presents as an example an overlay of seven partial spectra recorded for various values of the fraction w in the domain used for quantitative measurements.
In order to demonstrate the problems faced when the conventional simplified Kubelka-Munk Eq. 1 is applied, raw reflectance data are first divided by that of a ref- erence sample containing only DMB-treated silica. Nor- malization of measured spectra by that recorded for a 3-mm-thick layer of the standard dry KBr powder then yields absolute values of reflectance. Functions :Y(R) of both the relative and absolute reflectances are plotted in Fig. 4 against the mass fraction w of DMP.CN-deriva- tized silica present in the powder mixture. Linear plots passing through the origin, as predicted by Expressions 1, 5, and 18, are indeed not observed. Experimental re- sults obtained for absolute reflectance data exhibit a large intercept as well as a negative deviation from linearity with increasing absorbance. The problem due to the background absorption of the silica powder, which causes the intercept, is eliminated when reflectance measure- ments are referenced to that of a powder containing only DMB-treated oxide (w = 0). But in this case, even though the curve passes through the origin, a significant positive deviation from linearity is observed at higher concen- tration of the chromophore. In spite of many experi- mental precautions, which have proved to be sufficient in the case of ideal highly diluted systems, and which are described in the literature to give linear Kubelka-
APPLIED SPECTROSCOPY 1881
1.2 0.4
1.0 ~ , . , , , . , , , '" '"" 0.3
0.8 ~ , , m "'" 0 ~ -
~--- 0.6 0.2
T I , l , I ' " , I ,
E~, 0.4 - . . . . . . 0.1
0.2
0 0 0.2 0.4 0.6 0.8
w[-]
FzG. 4. Re la t i ve d i f fuse ref lectance, measured a t P = 2247 cm 1 fo r 3-mm-thick layers of mixtures of two LiChrosorb powders entirely covered by DMB and DMP.CN substituents, respectively. Simplified Kubelka-Munk functions of reflectance, normalized to that of the stan- dard 3-ram dry KBr layer (@), or normalized to that of a powder containing only DMB-covered silica (O), are plotted as a function of the weight fraction w of DMP.CN-derivatized powder in the mixture.
Munk plots, the present observations demons t ra te the inabili ty of Eq. 1 to describe satisfactorily diffuse reflec- tance measurements carried out with our derivat ized sil- ica powder mixtures.
Relative and absolute diffuse reflectance values, ex- t rac ted from the same exper imental data, are repor ted in Fig. 5 as a funct ion of the absorpt ion coefficient K of the cyano chromophore. Solid curves represent in bo th cases the best possible fit of Eq. 2 to the empirical points. The computer fitting procedure uses a mult ivariable non- l inea r l e a s t - s q u a r e s m e t h o d a d a p t e d f ro m Mar- quardt . 32,33 Since light t ransmission through the powder layer can be neglected and both the absorpt ion coefficient K and the layer thickness d are known, the only adjust- able pa ramete r here is the phenomenological scattering coefficient S of the sample. The poor correlat ion of the relative measurements with the curve calculated from Kubelka ' s hyperbolic model corresponds to the deviat ion from the l ineari ty already observed in Fig. 4. Th e best fit obta ined for absolute values looks even worse. With regard to these results, it appears therefore very tempt ing to t rea t diffuse reflectance da ta by normalizing them to the spec t rum of the DMB- t rea ted silica powder. This choice, however, can lead to mis in terpre ta t ion and errors, and excludes in practice a real de te rmina t ion of the scat- tering coefficient S. Indeed, the K u b e l k a - M u n k model, in its simplified or more complete forms, always yields nonlinear equat ions relating reflectance to the absorber concentrat ion. Only absolute diffuse reflectance should therefore be considered within this theory. This essential condit ion has been ignored or e luded in most published work, 9,34,35 where some authors even recommend in par- t icular cases the use of a reference made of a highly absorbent matrix26
To the exper imenta l points p lo t ted in Fig. 5b, one can apply simple corrections for background absorpt ion and Fresnel - type reflection. Correct ion factors /~R = (1 -- RF~ef')/(1 -- RF) and /~T = 1/(1 -- RF), suggested by Eqs. 7 and 8, require an evaluat ion of the Fresnel reflectance RF and RF ref" of the sample and of the KBr reference s tandard. Taking into account the difference existing
uJ o z k- o uJ =J LI. gJ rr
1.0
0.8
0.6
0.4
0.2
0 . 0 0
c c o b) o
A
0 0 0
i I , I , I , I ,
10 20 30 40 50
K CN [ cm -1 ]
FIG. 5. (a) Relative diffuse reflectance of LiChrosorb powders con- taining various concentrations of the CN chromophore. Values are normalized to that of a silica sample treated by DMB aminosilane only. (b) Absolute diffuse reflectance calculated from the same measure- ments at P = 2247 cm 1 and from reflectance of the standard dry KBr powder layer. Solid curves represent in both cases the best possible adjustment of Kubelka's hyperbolic model Eq. 2 to the experimental data.
between the densi ty of our loosely packed powders and tha t of the oxide bulk, the absorpt iv i ty a of the samples can be roughly es t imated from the absorpt ion coefficients K de te rmined earlier by the simple equation:
K" PSiO2 a ~ - - (19)
2- Ppowder "
Values for a obta ined in this way for Cab-O-Sil and LiChrosorb silica are approximate ly 30-40 cm -1. The concentra t ion of D M P . C N subst i tuents , for which the molar ext inct ion coefficient has been de te rmined as e = 27 mo1-1.L.cm -1, can be assessed from the surface con- centra t ion F = 3.9.10 -s mol /m 2 and the thickness of the organic functionalized layer, es t imated from van der Waals radii to 10/~. Th e upper l imit value found is c = 3.9 M, which corresponds almost exactly to the concen- t ra t ion of the pure liquid D M P . C N aminosilane. Ab- sorpt ivi ty a = e. c of these subst i tuents at ~ = 2247 cm -1 is then es t imated as equal to a = 105 cm -~. Corresponding imaginary refractive indexes K = a" (},/47r) lie therefore in the range between 10 .3 for silica substrates and 4.10 .3 for the chromophore layer. Such very small numbers appear negligible in comparison to the real par t of the powders ' refract ion index (n > 1). In the absence of light absorption, the mean external reflectance of a planar surface i l luminated with diffuse nonpolar ized incident radiat ion can be expressed by a relat ion recent ly estab- lished by Mandelis e t al.: 37
8 1 n 4
3n3 + 2 n 2 - 3 n s - 8n 6 + 6n 4 +1 R ~ = + 2(n 2 - 1)2 2(n 2 + 1)2(n 2 - 1)2
2n 3 8n4(n 4 + 1) + • In(n)
(n 2 + 1) 2 (n 2 + 1)(n 2 - 1)4
n2(n 2 - 1 ) 2 . 1 n ( n + 1~ (n 2 + 1) 3 \ n - 1]" (20)
If polarizat ion of incident light is neglected and the
1882 Volume 46, Number 12, 1992
T A B L E I lL Fresnel reflectance values at ~ = 2247 cm -~ (Rr) and cor- responding diffuse reflectance and transmittance correction factors (#t, #T) obtained for the dried KBr standard sample and both types of silica powders treated up to their surface saturation point by D M B and D M P . C N aminosilanes.
Sample RF fir /~r
Dry K B r s t anda rd 0.024 1 1.02 DMB-reac t ed LiChrosorb 0.106 1.09 1.12 DMP.CN- reac t ed LiChrosorb 0.075 1.06 1.08 DMB-reac ted Cab-O-Sil 0.074 1.05 1.08 DMP.CN- reac t ed Cab-O-Sil 0.063 1.04 1.07
refractive index of the powder particles is taken to be equal to n = 1.55 for KBr and n = 1.46 for silica, 3s Ex- pression 20 yields R~ values of 0.099 and 0.084 for planar surfaces of both materials, respectively. Experimental values of the Fresnel reflectance are determined at ~ = 2247 cm -1 by measuring the total collected light inten- sities I,i and I±, obtained by use of pairs of either aligned or crossed polarizers, respectively, and applying Eq. 11. These values, along with the diffuse reflectance and transmittance correction factors/3 are reported in Table III.
Measurements of the standard KBr powder's external reflectance are lower by a factor of four as compared to the calculated RF value. In the case of LiChrosorb and Cab-O-Sil samples though, the numbers are very close to those estimated by Eq. 20. This difference of behavior can probably be explained by a difference in the mor- phologies of the two powders, and the difference in ge- ometry on a microscopic scale of their exposed surface. The upper layer of derivatized silica samples is easily flattened. Brittle grains of LiChrosorb and very small Cab-O-Sil spherical particles under the action of the press piston form a smooth continuous plane. On the contrary, the KBr powder, which consists of larger angular gran- ules, is more likely to present asperities and faults on the scale of crystallites, which can cause multiple reflec- tions on their faces. Replacement of DMB substituents
0.5 W
< b- 0.4 o I.U ,-I t,I. I,U r r 0.3 UJ I-- (J
er 0.2
O
0.1 , I , I , I , I , 10 20 30 40 50 60
K t o t a l [ c m ' l ]
Fro. 6. Absolute diffuse reflectance of a - ram- th ick LiChrosorb powder samples , character ized by var ious concent ra t ions of CN chromophore . Global absorp t ion coefficients repor ted in the abscissa include back- g round absorpt ion due to the silica subs t ra te a t P = 2247 cm -1. Mea- s u r e m e n t s are corrected for Fresnel reflection by appl icat ion of factors fl descr ibed in the text. T he solid curve represen ts the bes t fit of Eq. 2 to the exper imen ta l points , wi thout invo lvement of any profile of optical proper t ies of the sample . F i t t ed sca t te r ing coefficient is S = 26.1 cm 1.
uJ O Z I,- 0 ILl ,,-I I / . ILl
I I I
4000 3500 3000 2500 2000
V [cm "1 ]
FIG. 7. Raw diffuse reflectance spec t ra of 1 .3-mm layers of Cab-O-Sil silica powders ent irely covered by D M B (A) and D M P . C N (B) siloxy subs t i tuen t s . N u m b e r of coadded scans: 250; resolution: 4 em-l; inter- fe rometer scann ing speed: 0.5 cm/s .
on the oxide surface by those of the DMP.CN aminosi- lane also seems to decrease the Fresnel reflectance. The observed attenuation of 25 % cannot be attributed to the negligible absorbance of a chromophore monolayer an- chored on the reflecting faces of directly exposed parti- cles. A better match of the refractive index of silica (n = 1.46) to that of the DMP.CN substituents (n = 1.453) compared to those of DMB (n = 1.428) might be at the origin of this effect. In the following, a linear interpo- lation has been used to evaluate reflectance and trans- mittance correction factors for any powder mixture con- taining a substituent molecular fraction y of the -C--N chromophore:
/~ = y.[/~ (DMP.CN) - /~ (DMB)] +/~ (DMB). (21)
Figure 6 shows data points obtained from measure- ments already presented in the two previous figures. In order to take background absorption into account, the abscissa is raised by a term corresponding to the silica absorption coefficient Kb (~ -- 2247 cm -1) = 13 cm -1. At the same time, the effect of Fresnel reflection is corrected for by the factor/3. This correction represents an increase of raw absolute diffuse reflectance data, ranging from 6 % at high chromophore concentration to 9 % for lower light absorption. The best fit of the hyperbolic Relation 12 to corrected experimental points is illustrated in the graph. The excellent correlation obtained shows that, with a few corrections to the measurements, Kubelka's model does apply in a satisfactory manner, without hav- ing necessarily to invoke a gradient of the phenomeno- logical parameters. The value of the scattering coefficient of derivatized LiChrosorb powder produced by the fitting procedure is established in these conditions as S = 26.2 ± 0.5 cm-L
The same kind of experiment is conducted with mixtures in various proportions of two Cab-O-Sil pow- ders, treated up to the saturation of their surface by aminosilanes DMB and DMP.CN, respectively, for which raw diffuse reflectance spectra are reported in Fig. 7. The absolute diffuse reflectance of 3-mm-thick samples is ex- tracted from spectra at the characteristic maximum ab- sorption of nitrile groups at ~ = 2247 cm-L Figure 8 exhibits the graph of the simplified Kubelka-Munk func- tion ~F(R), calculated from raw measurements of the ab-
APPLIED SPECTROSCOPY 1883
9
8 7 " - O ~
~re 6 . /
~ s /
s. 4 ,.
3 , , , , I , , , , I , ~ , ,
0 5 10 15 KCN [ cm -1 ]
Fro. 8. Simplified Kubelka-Munk function (Eq. 1) of absolute diffuse reflectance determined at ~ = 2247 cm 1 for 3-mm-thick Cab-O-Sil powder samples containing various concentrations of DMP.CN sub- stituents. Absorption coefficients K reported in the abscissa are cal- culated from the concentration of the chromophore only.
solute reflectance. The winding curve obta ined is indic- ative of the failure of the simplified model. Apar t from the silica substra te background absorption, which has not been taken into account, such a resul t can be ex- plained by the insufficient opacity of the powder layer. At low concentra t ions of the chromophore, part ial light t ransmission by the sample causes a decrease in the mea- sured diffuse reflectance relative to the model 's predic- tions, which were established on the basis of an infinite dep th hypothesis. The par t of light t r ansmi t t ed through the scattering layer diminishes when absorpt ion is in- creased. A be t te r l ineari ty is therefore slowly reached for coefficients exceeding K = 8 cm-L
After corrections for background absorpt ion and Fres- nel reflection have been applied to the same measure- ments , Kubelka 's complete hyperbolic model is brought into play. Global absorpt ion coefficients K ~ are cal- culated by adding to coefficients est imated from DMP.CN subs t i tuent concentra t ion a constant t e rm Kb = 3.0 cm -~, de te rmined earlier for the Cab-O-Sil silica support . Ab- solute diffuse reflectance is affected, moreover, to a 4 to 5 % extent by correct ion factors/3 given by Table III and the interpolat ion funct ion (Eq. 21). Corrected points ob- ta ined in these conditions are plot ted in Fig. 9, where the dot ted curve represents the best possible fit of Eq. 2 to the data. A systematic deviation of exper imental points relative to the calculated funct ion is observed, so tha t the quant i ta t ive theory appears in its tu rn as par- tially inadequate. Profiles of phenomenological absorp- t ion and scat ter ing coefficients K and S have therefore to be in t roduced in the model. The quasi-differential layer stack i terative procedure, presented above and summarized by Relations 16 and 17, is incorporated in the mult ivar ia te nonlinear least-squares fitting method. Several types of funct ion K =.f(z) and S = .f(z) have been tested. Best results are obta ined for exponential profiles of the coefficients in the dep th of the sample:
K ( z ) = C + B . e x p [ - A . ( 1 - z/d)] (22)
S ( z ) = C' + B ' . e x p [ - A ' . ( 1 - z/d)] . (23) When phenomenological parameters K and S remain
cons tant th roughout the ent ire sample layer, Kubelka 's model yields one expression. Since the scat ter ing coef-
,,, 0.16 ~ , ~ ~
0.14
0.12 , - I I , I .
0.10 ° W
0.08 n -
0.06 "---0---
0.04 , I ~ I , I , I ~ I , I
2 4 6 8 10 12 14 16 total [ cm -1 ]
FIG. 9. Absolute diffuse reflectance of Cab-O-Sil powder mixtures containing various concentrations of DMP.CN substituents as a func- tion of the global absorption coefficient. Measurements performed at
= 2247 cm -~ are corrected for Fresnel reflection. The dotted curve represents the best fit of Eq. 12 to the experimental points. The solid curve is calculated by the numerical iterative procedure after intro- duction of the K and S coefficients depth profiles of the sample. Best fit of the parameters in Relations 12, 22, and 23, simplified by equality C = C' = 0, yields in this case the values A = 0.80, A' = 0.34, and B' = 1.5cm 1.
ficient S is the only unknown paramete r in this case, the mathemat ica l system is ent irely de te rmined for each in- dividual exper imenta l point, and a unique solution is expected. In t roduct ion of an addit ional relation, consti- tu ted by the measuremen t of several points at various absorpt ion coefficients, allows us, therefore, to judge whether the model is appropriate . Taking into account inhomogeneit ies of the K and S coefficients introduces a number of addi t ional unknown quantit ies. Apar t f rom the model and both profile equations, expressing the mean absorpt ion coefficient/~, repor ted along the ab- scissa, yields an addit ional relat ion tha t limits the num- ber of unde te rmined parameters in Eq. 22 to 2, and limits to 5 the total number of variables tha t have to be adjusted simultaneously. Since under these condit ions the math- ematical system still remains doubly undetermined , a s impl i fca t ion of Expressions 22 and 23 is necessary.
By arbi t rar i ly sett ing C = C' = 0, one causes the com- puter ad jus tment of all o ther unknown quant i t ies of the model to corrected exper imenta l da ta to lead to the val- ues: A = +0.80, A' = +0.34, and B' = 1.5 cm -1. The last two parameters assess the mean scattering coefficient of derivat ized Cab-O-Sil samples as S = 1.25 cm-L The solid curve t raced in Fig. 9 represents , under these conditions, the numerical ly evaluated absolute diffuse reflectance dependence of the mean absorpt ion coefficient. The per- fect fit of exper imenta l points by the theoret ical curve demonstra tes , thanks to the addit ional in t roduct ion of profiles of the phenomenological coefficients in the dep th of the layer, the final success of the general Kube lka - Munk model. A change in the order of magni tude of +_ 5 % relative to opt imal parameters induces a noticeable de- ter iorat ion of the qual i ty of the fit. Nothing, however, prevents the use of other types of profile equations, which, along with another dis t inct set of parameters , may lead to a fit of comparable quality.
For bo th absorpt ion and scattering, the positive values of A and A' parameters t ha t have been obta ined imply a decrease in the coefficients with increasing dep th of
1884 Volume 46, Number 12, 1992
0.5 W {.} Z < 0.4
if} =, 0.3 <
I -
0.2 tU
,-I <
0.1
0 Z
0 4000
B /
3500 3000
[cm -1 ]
_e
I
2500 2000
FIG. 10. Normalized diffuse transmittance spectra of 1.3-mm-thick samples of Cab-O-Sil powders entirely covered by DMB (A) and DMP.CN (B) substituents. Photopyroelectric signals normalized to the spectrum of the blackened PVDF sensitive element yield absolute transmittance values. Number of coadded scans: 75; resolution: 4 cm l; interferometer scanning speed: 0.03 cm/s.
the sample. Such a larger light absorption at the layer's surface by a factor of about two, relative to that char- acterizing the bottom of the sample, can be simply ex- plained by the powder compression produced during its leveling by the press piston. For LiChrosorb powders, which are denser and much less compressible than the very loose Cab-O-Sil aerogels, the same leveling opera- tion did not appear to induce the inhomogeneities of the optical coefficients. Although the scattering coefficient S is affected to a smaller extent, results obtained for the adjustment of parameter A' would indicate that this one is larger by ~20% at the surface in comparison to its mean value throughout the whole sample. In a simplistic approach, one can consider the scattering parameter S as a measure of the number of interfaces encountered by light per depth unit. Under packing, an increase in the density of a powder consisting of relatively distant par- ticles should in a first stage increase, as well as the con- centration of scattering centers. More detailed studies of the effect of compression of our powders on the phe- nomenological scattering coefficient are reported else- where. 21
Kubelka-Munk theory, adapted to the nonideality of our systems, has been tested so far only for diffuse re- flectance description. General hyperbolic solutions 2 and 3, though, allow us to consider simultaneously the diffuse transmittance of the samples. Layers 1.3-mm thick, con- stituted of derivatized Cab-O-Sil mixtures, are prepared in the photopyroelectric detector/sample holder where 10 mg of the silica powder is typically homogenized by vibrations. Diffuse reflectance and transmittance spectra are recorded consecutively without any modification of the sample and of the optical geometry between both measurements. Figure 10 shows, in particular, diffuse transmittance spectra obtained from the same powder layers used for recording diffuse reflectance spectra of Fig. 7. After normalization by reference spectra obtained for the standard 3-mm-thick, dry, KBr sample and the blackened PVDF film, respectively, reflectance and transmittance data obtained at ~ = 2247 cm -~ for various concentrations of the C-= N chromophore are corrected for background absorption of the silica substrate and the
1.0
0.8 i -
a 0.6 IJJ k - o IJU m 0.4
O O
0.2
0.0
m
~ , I , I ~ I , I , I ,
2 4 6 8 10 12 14
K total [ cm -1 ]
FIG. 11. Absolute diffuse reflectance and transmittance of 1.3-mm layers of mixtures containing various proportions of Cab-O-Sil powders covered by DMB and DMP.CN substituents, respectively. Measure- ments recorded at p = 2247 cm -~ as a function of the global absorption coefficient are corrected for Fresnel reflection. Solid curves represent the best possible fit of Eqs. 12 and 13 to the experimental data, cor- responding to a scattering coefficient S = 1.10 cm ~.
Fresnel reflection. Absolute diffuse reflectance and trans- mittance values produced in these conditions are plotted in Fig. 11. As for previous data, the order of magnitude of experimental errors is estimated at approximately +5%. Kubelka's hyperbolic solution Equations 2 and 3 are simultaneously adjusted to both sets of experimental points by the computerized nonlinear fitting procedure. Both reflectance and transmittance curves correspond- ing to optimally adjusted common parameters coincide in a very satisfactory manner with their respective data. Since it is very unlikely to occur in such a thin scattering layer--on which, moreover, no mechanical pressure has been applied--the dependence of optical properties on the depth of the samples does not need to be invoked here. Since the phenomenological scattering coefficient under these conditions represents the only unknown pa- rameter of an entirely determined system, its adjustment yields a value of S = 1.10 cm -1. In thicker layers of the same Cab-O-Sil powders for which inhomogeneities in the optical properties could not be neglected, the scat- tering coefficient characterizing the bottom of the sample (z = 0), where the effect of compression is minimal, has been determined as S(0) = B ' . e x p ( - A ' ) = 1.08 cm -1. The remarkable coincidence of this last number with that of the scattering coefficient obtained for simultaneous adjustment of diffuse functions of thin layers nicely con- firms the general validity of quantitative models and procedures employed.
CONCLUSIONS
Kubelka's general equations are used to describe quan- titative diffuse reflectance and transmittance measure- ments on powders. Silica samples with a well-defined amount of covalently bound surface chromophore serve as model systems. Comparison of FT-IR measurements of diffuse reflectance and transmittance with model cal- culations indicates the necessity of correcting for the bulk absorption of the silica powders and their specular reflection. The introduction of inhomogeneous optical parameters that vary with the depth of the packed pow-
APPLIED SPECTROSCOPY 1885
der l ayer f u r t h e r i m p r o v e s a g r e e m e n t b e t w e e n exper i - m e n t a n d theory .
Bes ides , t he a b s o l u t e s c a t t e r i n g coeff ic ients of Cab -O- Si l a n d L i C h r o s o r b s i l ica h a v e b e e n d e t e r m i n e d in t he i n f r a r e d reg ion a n d r e l a t e d to t he m o r p h o l o g y of t he powders . T h e a p p l i e d m e t h o d a p p e a r s v e r y p r o m i s i n g for a t r u l y q u a n t i t a t i v e a p p l i c a t i o n o f F T - I R spec t ros - copy to n o n i d e a l p o w d e r s a n d rough surfaces .
ACKNOWLEDGMENTS
The authors would like to thank Drs. Ph. Schneider and R. Cloux for the preparation of the aminosilane reagents. Valuable advice and help from Professor E. sz. Kovfits and Dr. L. Jelinek in the preparation and characterization of surface derivatized silica are gratefully ac- knowledged. Financial support of this work has been provided by the Fonds National Suisse de la Recherche Scientifique.
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