Statistical analysis of non-additivity in spectra of multicomponent mixtures

R

~~~a~ Research Paper

281

Chemometrics and IntelligentLaboratorySystems, 18 (1993) 281-284 Elsevier Science Publishers B.V., Amsterdam

Statistical analysis of non-additivity in spectra of multicomponent mixtures MS. Khots, V.I. Nazarov and A.A. Lyovin Ail-Russian Scientific Research Institute of Oil Refining, Aviamotornaya 6, 111116 Moscow (Russia)

(Received 9 July 1991; accepted 12 June 1992)

Khots, MS., Nazarov, V.I. and Lyovin, A.A., 1993. Statistical analysis of non-additivity in spectra of muiticomponent C~~o~etr~cs and intelligentaerator Systems, 18: 281-284.

mixtures.

Simplex-lattice designs are used to evaluate non-additivity in the spectra of multicomponent mixtures. Interactions of the components, which result in deviations from additivity, are shown. An algorithm is presented to determine a composition region in which violation of the additivity is revealed to the greatest extent. The method described can be used for the spectral analysis of mixtures, such as the products of petrochemical synthesis.

INTRODUCTION

Spectroscopy has been used for the quantitative analysis of multicomponent mixtures for many years. Such analyses are sometimes complicated due to violation of the spectral additivity. Such violations can result, for example, from the chemical interaction of the components. In this case more reference standards are needed for the analysis to evaluate the component interaction effects, to plot rather complicated calibration relationships, etc. It has been noted that consider-

Co~e~~~nce to: Dr. MS. Khots, All-Russian Scientific Research Institute of Oil Refining, 111116 Moscow, Aviamotornaya 6, Russia. Present address: Farmaceutisch Instituut, Vrije Universiteit Brussel, Laarbeeklaan 103, B-1099 Brussels, Belgium. 0169-7439/93/$06.00

able distortions in the results of a spectral analysis can occur if such violations of spectral additivity are neglected. In this connection the development of a method for the statistical appraisal of the non-additivity properties of the spectra of multicomponent mixtures seems to be urgently required.

THEORY

Assume that Eik, Ez’ are spectra of the components and mixtures, respectively, where E,, , EgiX are specific factors of e~inction, i = 1,. . . , n; n is number of ~om~nents; j = 1,. . . , N, N is number of mixtures; k = I,. . . , K, K is number of points in the spectra used for the analysis.

0 1993 - Elsevier Science Publishers B.V. All rights reserved

MS. Khots et al. / Chemom. Intell. Lab. Syst. 18 (1993) 281-284

282

If the additivi~ criterion is satisfied, then with an accuracy up to a random error

where cii is concentration of the ith component in the jth mixture (with an accuracy up to sensitivity factors). Since for concentration the relation &cjj=

1 for

j= l,...,N

is valid, the additivity can be checked by the methods of experimental design, using a composition-property diagram. The mathematical design of an experiment is usually conducted in points {n, q} of the lattice in the simplex, where n is the number of components, and q is the degree of a polynomial [l]. It is natural to assume that the characteristics of the spectral lines of mixtures represent a rather good, smooth function of component concentrations, which can be approximated by a polynomial of a low degree (4 G 3). The selection of the degree of the polynomial in an experimental design is determined by the number of components and by experimental error, i.e. by specific conditions of the task under consideration. After conducting the experiments a linear model (ML,,) is plotted and its agreement with the experimental data is checked. For that purpose a residual sum of squares Sz,, for each point of the spectra being considered is calculated: 2

N

Ej”k”- i

&= j-1 ~[

cijEik

i=l

The following relationship

I

is then considered:

% Fobs =

f L d

where fL = N - it is the number of degrees of freedom of the linear model, si, and k = 1,. . . , K is the deviation of a specific extinction factor in the kth experimental point. Fobs is compared to a critical value determined by the F distribution table for the confidence level of probability p. If a linear model agrees

/ Original Research Paper

8

with the experimental data it means that the additivity for the given spectral point is satisfied with an accuracy up to experimental error. Otherwise successively paired factor interactions fir, i,> are entered into the linear model and the compliance of the models obtained (Mil,i2,k), i,, i, = 1,. . . , n, i, #i, is checked against the experimental data. If any of the models Mi,,i2,k describes the experimental data, then the appropriate paired interactions can be considered, from the chemical standpoint, as elements of the mathematical model, indicating interactions (chemical or intermolecular) of the components of the mixture. If models incorporating paired interactions do not agree with the experimental data, then one has to consider additional members describing triple interactions, i.e. Mi,,iz i,k, i,, i,, i, = l,..., n, i,#i,#i,,etc. ” After the formation of a set of satisfactory mathematical models a region of composition variations can be determined in which deviations from the additivity criterion are at a maximum for each experimental spectral point. To this end the difference Mi,,i,,._,,k - M,, can be considered and its extremum evaluated in terms of (c,, . . . , c,). A value of extr(M,,,i *,,,,, k - M,,,) < 0 indicates an antagonistic effect of the component interaction; if extr(Mi,,i,, . ,. ,k - ML,,) > 0, synergism is observed. RESULTS AND DISCUSSION

The following complex compounds were used: (A> a shale distillate atmospheric residue containing phenolic and car-bony1 compounds; 03) is a mixture of sulfonates; and (C) is a mixture of dithiophosphates. The selection of the components was determined by the following factors: (1) It is known a priori that rather strong intermolecular interactions are observed among the components in the mixture [2]. (2) Components A, B and C are used in the development of effective corrosion inhibitors, To evaluate the spectral interactions of the components a simplex-lattice design was implemented (Fig. 1). The spectra were recorded on a Perkin-Elmer 457 IR spectrometer in the 4000-

H

MS. Khots et al. / Chemom. Intell. Lab. Syst. 18 (1993) 281-284

/ Original Research Paper

283

TABLE 1 Values of the F statistic for different models for a segment of the OH group band * v (cm-‘)

Number of points k

1-o

A

0

0,2

O,L

0.6

0,6

1

c

Fig. 1. Simplex-lattice design to evaluate spectral interactions.

700 cm- * region using NaCl cuvettes of 0.0025 cm layer thickness. Optical densities were measured in the IR spectra of the mixture, corrected to 100% component content, divided by the width of the cuvette layer. The resulting specific extinction factors were calculated for each band; the transformation of the spectrum from the analogue form to the numerical form was carried out using a halfwidth of the band after each lo-20 cm-‘. Eight-three different points in the spectra were considered. The study of the IR spectra of the mixture ingredients allowed the determination of the bands which may be of interest. There are bands at 3520-3240 cm-’ (OH group), 1700-1680 cm-’ tC=O group), 1620-1540 cm-’ (C=C bond in the aromatic ring), 1240-1140 cm-’ (sulfa group), 1040-970 cm-’ (P-O group), 860-790 cm-’ (deformation vibrations of SO, group), and 670-640 cm- ’ (deformation vibrations of COP group). It may be noted that the first three bands are the most typical for component A, the fourth and fifth are typical of C, and the sixth is typical of B. The linear models were plotted first in all cases. It was found that in some cases they did not agree with the experimental data. Satisfactory models were obtained after paired interactions were entered into the models. Tables 1 and 2 show the values of the F statistic as an example. Values are shown which were obtained for linear models, as well as those comprising various paired interactions (for a segment of the OH group band) and for the model a B-C interaction (for

10 11 12 13 14 15

3340 3320 3300 3280 3260 3240

s: k

2 12s:

GBk & 11s:

tsic, 11s:

;Gc, 11s;

1.15 2.65 2.16 2.36 3.00 3.66

1.20 1.85 2.28 2.58 2.98 3.82

1.13 0.89 0.87 0.72 1.72 1.08

1.68 2.48 2.15 2.43 2.89 3.51

* F&, (0.99, 12, m)= 2.18. F,ri, (0.99, 11, m)= 2.24.

the P-O group band); sk values were equal to 7-8% of the specific extinction factor [3]. Analysis of estimated data allowed the conclusion to be drawn that the inclusion in the linear model of those terms characterizing paired interactions of the com~nents can result in an increase of the F relationship in comparison to linear models (Table 1). It is worth noting that the largest deviation from additivity (maximum of the F relationships may not coincide with the band ma~mum. Thus, on the P-O group band the F relationship maximum is observed at v = 980 cm-’ (Table 21, whereas the actual band maximum is found at v = 1000 cm-’ (Fig. 21. As a rule, for different bands various paired interactions had to be included in the linear models in order to obtain mathematical models that provided an adequate explanation of the experimental data. Thus, the formation of a satisTABLE 2 Values of the F statistic for linear models and for models comprising B-C interactions for the P-O group band Number of points

v (cm-‘)

15: k 12s:

& 11s;

1040 1030 1020 1010 1000 990 980 970

2.16 1.63 1.98 2.62 1.96 2.40 4.73 1.77

1.94 1.24 1.10 1.51 1.36 1.40 1.26 1.52

k

40 41 42 43 44 45 46 47

‘

284

M.S. Khots et al. / Chemom. Intell. Lab. Syst. 18 (1993) 281-284 / Original Research Paper

factory mathematical model on the basis of the OH group bands was associated with the inclusion of the A-C paired interaction; i.e. OH groups of shale distillate atmospheric residue (component A) interact with al~ldithiophosphates (component 0, On the P-O band a paired interaction between the P-O group of al~ldithiophosphates (component C) with sulfogroups (component B) is revealed. In the bands of the aromatic C==Cbond, the C=C group in the phenolic compounds of shale atmospheric residue (component A) interacts with sulfonate sulfogroups (component B). The results of our calculations have demonstrated the well known spectroscopic postulates of potential intermolecular interactions [2] and have revealed the most signi~cant of them for the triple system under study. It should be mentioned that, in most cases, the linear models adequately describe the experimental data. Thus, in the experiment under consideration, the spectra of the mixtures show additivity at many points, and a quantitative spectral analysis can be carried out in the usual way; i.e. neglecting the interactions of the components. In our opinion the study of the points in the spectrum where violation of the additivity criterion is at a m~mum is of interest. Therefore it is useful to consider the difference between an adequate model comprising paired interactions and a linear model and to test its extremum on the composition-property plots. Fig. 3 shows the function level lines

A.0

0.2

o,4

0,6

0.8

Fig. 3. A(cA, ca, cc) = Ma,c,ti -ML,,

n

1 c level lines.

for the band of the P-O group at the point v = 980 cm-‘. The area of maximum violation of additivity adjoins the B-C edge. It is limited by the line of A(c,, cu, co) = -20 level. The minimum of A(c,, cn, c,) is naturally on the B-C edge. A negative sign of A(c,, ca, co) means that in this case there is an antagonistic effect in the interaction between components B and C. Analysis of the diagram permits us to conclude that, even in this point of the spectrum (V = 980 cm-“), a large area of compositions is observed, in which the additive property is satisfied (the vicinity of the line of A(c,, cu, co) = 0 level).

CONCLUSION

A method has been presented for the evaluation of non-additivity in the spectra of multicomponent mixtures. The method permits one to single out areas of composition where deviations from additivity are clearly seen. In the course of the quantitative spectral analysis of mixtures in these areas, the interactions of ~m~nents should be taken into consideration.

A@,, ca, cc) = M,,,,M - ML,~G

5 i:

e

REFERENCES

Fig. 2. P-O bands for 15 samples of A-B-C

1 H. Scheffe, Simplex-lattice design, Journal of the Royal Stat~t~~l Society, Ser. B, 20 (1958) 344. 2 C, Reichardt, Losungsmittel-Eflekte in der organixhen Chemie, Verlag Chemie, Berlin, 1969. 3 W. West (Editor), Chemical Applicafions of S~ctroscopy, Interscience, New York, 1956.

s 8

-7itiisc--.

1100 1000

1000

:

1100 1000 000

mixtures.

cm-’