Anal. Chem. 1991, 63, 1685-1691
of system parameters, rather than by the trial-and-error approach adopted in this study. Being completely computer controllable and compatible, SIA should lend itself ideally to the simplex optimization studied.
(6) Ruzlcka, J.; Marshall, G. D.; CMsUan, 0. D. Anal. Chem. 1000, 6 2 ,
1861. (7) American Conference of Governmental Industrlal Hygienists. We&old Llmlt Values and Bbkglcai Exposure Indlces; Clnclnnatl, OH, 1988. (8) Well. T.; Ruzicka, J.; Christian, G. D. Telenta, In press. (9) Z a b u , K.; Yamaglshi, K.; Ohkura, Y. Chem. phann. Bull. 1088, 36 (1I), 4488. (10) Kelly, T. A.; Christian, G. D. Anal. Chem. 1082, 54. 1444. (11) Kenaoka. Y.; Takahashl, T.; Nakayama. H.; Tanlzawa, K. Chem. phann. BUN. 1085, 33(4), 1721.
ACKNOWLEDGMENT Cecille Grataloup's technical assistance is gratefully acknowledged along with G. D. Christian's kind interest in this project. Registry No. Subtilisin, 9014-01-1. LITERATURE CITED Ruzlcka, J.; Hensen, E. H. Fbw In/ecmOn A n a m , 2nd ed.: Wlley 8 Sons: New York, 1988. Rurlcka, J.: Marshall, G. Anal. Chlm. Acta 1090, 237. 329. Hungerford, J. M.; Christian, G. D.; Rurlcka, J.; Glddlngs. J. c. A M / . Chem. 1085, 5 7 , 1794. Masoom, M.; Wwsfokl, P. J. Anal. Chlm. Acm 1086, 179, 217. Monde. H. A. K ~ H C o ~ ~ ~ ~wileyla Sons: ~ New York, 1988.
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RECEIVED for review February 20,1991. Accepted May 30, 1991. We express our gratitude to Novo-Nordisk, Denmark, for supporting this project by purchasing necessary materials and reagents and to the National Science Foundation of Switzerland for funding of T.G. during his stay a t the University of Washington, as well as to the National Institute of General Services (Grant No. SSS-6 (b) 1R43 GM b y Medical ; 45087-01) for partial financial assistance.
Factor Analysis of the Thermogravimetry of Rubber Blends by Singular Value Decomposition and Variance Minimization C. H.Lochmiiller* and S. J. Breiner Department of Chemistry, Duke University, Durham, North Carolina 27706 M. N. Koel Institute of Chemistry, Academy of Sciences of the Republic of Estonia, Tallinn, Estonia, USSR 200026
M . A. Elomaa Department of Polymer Chemistry, University of Helsinki, Meritullinkatu l A , SF-001 70 Helsinki, Finland
Thermogravlmetry of a wries of rubber blends and mixtures of chkroprene rubber (CR), MclClkneacryknlbRe rubber (NBR), and common rubber addnlves was performed under fbwlng He from 200 to 550 OC/mIn. The resuits were analyzed by udng factor analytical methods. Singular value decompodilon of the derivative data (DTG) matrix Indicates thai, upon bbnding wlth even small a"tsof NBR, CR loses most of its dlsiinctlve thermal character, leaving a blend conslstlng partially of native NBR and a composite species unlike either of the two siartlng materials. Removal of NBR and CR Influences from the curves was accomplished by udng a constrained variance mlnlmlzatlon procedure. The r e d i s are consistent W h an argumeni that new species are generated during blending.
INTRODUCTION Applications of thermogravimetry (TG) are quite varied, due, in part, to the conceptually simple nature of such determinations. The resulting data are correspondingly simple, though the processes leading to the changes in mass of the sample may be extremely complicated. It is in the interest of economy to limit this introduction to the approaches common to elastomers and especially to rubbers. Practitioners of TG in the rubber industries have found many uses for it including the determination of thermal 0003-2700/91/0363-1685$02.50/0
properties, investigations of reaction kinetics (e.g., ref 11, and quantitation in systems where the composition is approximately known and an assay is desired. Typical quantitative uses include the determination of moisture, volatile organics, oil, filler, and plasticizer concentrations (2). In such determinations, the material is typically heated at a preprogrammed rate (ca 20-150 OC/min) and the desired quantity is calculated from the mass loss corresponding to the pyrolysis (or evaporation) of the component of interest. This approach is recognized as simple and reasonably accurate, and the use of TG in compositional determination recently has been the object of standardization by the ASTM (3). Certain determinations, such as that for carbon black, depend on the ability to modify the atmosphere during the TG run (e.g., refs 4 and 5). A persistent problem in TG and particularly with polymer samples is reproducibility. The many known influence on TG response and the difficulties with their control are reviewed elsewhere (6-9). The use of samples finely divided with a cutting blade seems to alleviate many of these problems (IO). In the work reported here, sample sizes of 1-2 mg were found to be optimal for reproducing TG features. The question addressed in this paper is whether TG curves can provide more than simple, bulk-compositional determinations-i.e., whether information about structural details of the elastomers is reflected in such curves. If TG curves do contain such information, it might be revealed when the results are viewed by using multivariate techniques in the proper context. If, for example, one is interested in subtle 0 1991 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 17, SEPTEMBER 1, 1991
Table I. Rubbers and Blends Used in the Study (Compositions in Parts by Weight) 1:0 NBRb NBR (ACN = 36%)
8:2 N8C2bb
67.48 0.00 26.99 2.70 0.67 0.74 0.13 1.28 0.00 99.99
54.00 13.50 N-550 carbon black 27.00 ZnO 2.70 stearic acid 0.67 CBS" 0.59 diphenylguanidine 0.11 sulfur 1.03 0.40 MgO total 100.00 N-Cyclohexyl-2-benzodiazosulfenamide. Label in figures. CR
NBR/CR ratio 6 4 N6C4bb 4:6 N4C6bb 40.51 27.01 27.01 2.70 0.68 0.45 0.08 0.77 0.81 100.02
2:s N2C8bb
0 1 CRb
13.51 54.04 27.02 2.70 0.68 0.15 0.03 0.26 1.62 100.01
0.00 67.57 27.03 2.70 0.68 0.00
27.01 40.52 27.01 2.70 0.68 0.3 0.05 0.51 1.22 100.00
0.00
0.00 2.03 100.01
Table 11. Compositions of Mechanical Mixtures of NBR and CR
NBR, r g CR, r g % %
NBR CR
N8C2m
N6C4m
N4C6m
N2C8m
944 234 20 80
686 462 60 40
427
255 965
643 40 60
21
79
differences between otherwise similar rubbers, the derivatives of TG curves, which tend to emphasize changes in the signal, might best be used a t the expense of magnifying high-frequency noise. This paper reports the results of an investigation of the TG of butadiene-acrylonitrile rubber (NBR) and chloroprene rubber (CR) blends by application of singular value decomposition (SVD)analysis and an approach to matrix deconvolution similar to the method of rank annihilation (RA) with incomplete information (II).
EXPERIMENTAL SECTION Materials. All sampleswere aliquots of commercially prepared laboratory sample sheets. Each rubber sheet was formulated with nearly identical proportions of carbon black, ZnO, and stearic acid (octadecanoic acid: H3C(CHz)&OOH). Relative amounts of other additives varied linearly in proportion to blend components-e.g., CR. The rubber composition and additive amounts appear in Table I. The use of helium purge gas was indicated by the need to compare the TG studies with other pyrolysis experimentscarried out independent of the analysis presented here. The use of He as a purge creates some problems due to its high thermal conductivity and the changes in that property in the atmosphere of the furnace during pyrolysis. Each sample for TG was prepared by slicing thin strips approximately6 mm X 2 mm X 0.2 mm from a freshly cut surface of sheet rubber by using a methanol-washed razor blade. Small cuts (roughly2 mm X 0.2 mm X 0.1 mm) were taken from these strips until samples of approximately 1-2 mg were obtained. These were then stored in tightly capped glass vials until needed. At determination time, the platinum balance pan was cleaned by rinsing with methanol and subsequently heated to 950 O C and exposed to air to burn off volatile contaminants. Samples were prepared for TG by placing the appropriate mass of material in the pan, taking care to maintain as much separation as possible between pieces. Duplicate runs were made of each rubber and blend, and an additional series of sampleswas run, which consisted of mechanical mixtures of the two unblended rubbers by mass for comparison with the blended samples. The mass ratios of NBR to CR were adjusted to coincide with the blend compositions (Table 11). Instrumental Procedures. All TG curves were obtained on Perkin-Elmer TGA-7 running TAS7 software. The furnace was calibrated by using Curie point standards (Alumel, T,= 163 OC; and Perkalloy, T,= 596 "C) (12). The thermobalance mass scale was calibrated by using a Class M 0.100-g mass. Dry helium gas was employed as the balance purge at a flow rate of 40 mL/min and as the sample purge at 20 mL/min. Baseline correction was
,
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performed on every run. There is reason to be concerned that small sample sizes can lead to sampling problems in terms of sample homogeneityfor elastomer formulation studies. By using the methods employed in this study, thermogravimetry on replicate samples 6 months apart shows excellent agreement of the temperature of maximum mass loss;for example, for chloroprene rubber, the point of maximum mass loss rate was 302.3 "C with a standard deviation of 2.0 OC for three runs. This is well within the precision expected for Curie point calibration, i.e., 3.6 OC (13, 14).
Each sample was held at 100 OC for 1min, and then the mass was recorded. As a precaution to eliminate air from the furance, the furnace tube was purged for at least 5 min prior to the beginning of a sample run. The sample was then heated ballistically to 200 OC at which point the temperature ramp and data collection were started. All sampleswere analyzed over a temperature range of 200-550 "C at a ramp rate of 5 OC/min. After collection, the data were transferred to another computer and analyzed. The data for each run were treated by using a locally written program incorporating the smoothing and derivative generating routines after the method of Savitsky and Golay (15,16). Each TG data set consisted of at least 1600 data points,and derivatives were calculated by using a five-point, quadratic first-order derivative filter. A data point window of this size did not significantly distort the positions or shapes of features in the derivative TG (DTG) curves. As noted by Newkirk and Simons (13,DTG presents the same information as TG in a form that tends to emphasize the changes which occur during the TG experiment.
RESULTS AND DISCUSSION The T G curves, all normalized to the fraction of sample mass, for the various blended and mechanically mixed rubbers appear in Figures 1and 2. The curves for the blended rubbers are the averages of duplicate runs for each composition. Curves for the unblended rubbers (NBR and CR) also appear in each figure for reference. Although one can detect significant differences between the curves within each of the two
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Flgure 2. Integral TO curves for the pure and mechanically mixed rubbers. Labels correspond to those In Tables I and 11.
Flgure 4. Derivative TG curves for the pure and mechanically mixed rubbers. Labels correspond to those in Tables I and 11.
of a complex of CR and NBR that has thermal properties quite different than either of the unblended materials. Put another way, upon blending, essentially all traces of the native CR disappear. Data Analysis via Singular Value Decomposition. SVD Algorithm. The singular value decomposition (SVD) algorithm is a very powerful method for calculating seta of eigenvectors for the row and column spaces of virtually any given data matrix (18,19). The algorithm, briefly described, decomposes a data matrix, [D], into three compatible matrices as follows:
s
t
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Figure 3. Derhrathre TO curves for the pure and commercially blended rubbers. Labels correspond to those In Table I.
sets, a much more obvious comparison appears in the DTG curves, shown in Figures 3 and 4 (Figures 1-4 are plotted by using an evenly spaced subset of the complete data set). The focus of this paper is the examination of such differences, and what follows refers to the DTG view of the data obtained. The most distinctive feature in the DTG curve for pure CR is a "spike" at about 303 "C. We have obtained pyrolysis-masa spectral evidence that this corresponds to the rapid loss of HC1 from the rubber. It is interesting to note that no such feature appears in any of the TG curves for the blended rubbers (Figure 3) yet is present in all curves for the mechanical mixtures (Figure 4). The pyrolysis-mass spectral evidence for the blends still shows loss of chlorine but not the rate exhibited by CR at 303 OC. The major feature for unblended NBR is as a broad peak peaking a t about 448 O C . This feature appears in each sample containing NBR, blended or mixed. The third major feature which appears is the peak occurring at about 360 "C, which is most prominent in the curve for the blend of NBR and CR in proportions 2:8. The same feature is present in all of the blends, its height being roughly proportional to the fraction of CR originally present. The complete disappearance of the characteristic CR peak from the blend curves is, we believe, indicative of the final disposition of the CR in the blended rubbers. The blending process involves application of heat and intimate mixing of the polymers and additives. This apparently alters the structure of CR, perhaps cross-linking it with other components. The excess NBR apparently retains its individual identity, even within the blend microstructure. One may envision the blend as consisting of two interpenetrating continuous phases, one consisting of NBR and the other consisting
mxn
[VI [SI WIT mxp pxp pxn
where p is the rank of matrix, the superscript T denotes the matrix transpose, and the data below matrices give the matrix dimensions. As written, [U]is composed of column vectors that form a set of orthonormal eigenvectors for [DIT[D]. Similarly, [VI consists of columns (rows of [VIT)that form an orthonormal set of eigenvectors of [D][DIT. The matrix [SIcontains on ita diagonal the singular values of [D], which are equivalent to the square roots of the eigenvalues of the matrix [DIT[D](20). The SVD process may be thought of as a mathematical separation of the influence of the row designees ([VI), the import of each influence (the diagonal elements of [SI), and the corresponding influence of the column designees ([VI). It is important to bear in mind that SVD is a linear decomposition. Despite the fact that virtually any real matrix may be decomposed, if the "true model" for data is not linearly separable, the resulting decomposition is a purely mathematical construct, perhaps devoid of any physical significance. SVD Results. The SVD of the DTG curves for the rubbers, blends, and mixes shows some important relationships. Plots of the "factor spectra" related to the temperature designees in the data matrix appear in Figure 5. These curves represent the three largest eigenvectorsassociated with mass change as a function of temperature (and hence time) and are weighted by the corresponding eigenvalue to reflect the relative contribution toward the entire data set. By comparing these factor spectra with the curves for the plain rubbers (Figures 3 and 41, it is apparent that the first-row-related eigenvector consists predominantly of a signal corresponding to that of NBR. There are also small amounts of signal corresponding to the features associated with CR and our proposed composite material. This mixing of various signals is common with h e a r decomposition methods; the eigenvectors are constructed as linear combinations of the original data, and since the original vectors (TG curves) are not strictly orthogonal, the eigen-
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 17, SEPTEMBER 1, 1991
Table 111. Values of Functions Used in Determining the Number of Important Factors
no. of factors
eigenvalue
1 2 3 4 5 6 7 8 9 10
72.36 9.61 4.17 0.47 0.14 0.10 0.04 0.02 0.02 0.01
See ref
1
21.
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22.
See ref
%
variation explained 83.24 94.29 99.09 99.62 99.79 99.90 99.94 99.97 99.99 100.00
Malinowski’s indicator function
reduced eigenvalue
(~103).
(X102)b
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1.24 0.97 0.54 0.51 0.61 0.72 1.12 2.23 6.08
4.52 0.67 0.33 0.04 0.01 0.01 0.01 0.01 0.01
1.02 0.64 0.28 0.20 0.18 0.13 0.11 0.11 0.07
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vectors contain contributions from every signal in the data set. We consider only three factors (or eigenvectors) in the remainder of this paper. This decision is based on several of those criteria recognized in the literature as means of determining the number of significant factors in a given data set (21-23). Table I11 summarizes the results of these testa, each indicating that three or four factors sufficiently account for the variation in the data. The second eigenvector consists primarily of signal related to the TG curve for unblended CR. The third eigenvector is of more interest here since it contains a feature unrelated to either of the unblended rubbers. This vector is of particular interest in relation to the peak at 360 “C and its origin as proposed above. Worthy of note are the opposite signs of the central lobe to those of the two outer lobes. These closely resemble the characteristic shapes of the pure NBR and CR. This observation reinforces the conjecture that the material that decomposes around 360 OC is the result of a component which was not present in the preblended raw materials. The column factors, which can be viewed as the relative weights of the associated row factors to the TGs, show a correspondence with the row factors in terms of out interpretation of the SVD. Figure 6A shows that the magnitude of first factor is smallest for pure CR,increasing monotonically through the blends to a maximum for NBR. The same factor in the series of mixed samples shows a likewise monotonic increase but appears to have a nearly linear increase in value with increasing amounts of NBR. The series of blends, in contrast, displays significant irregularity in the progression. The pattern of column factor 2 is more interesting yet. The influence of this factor is clearly very important to the DTG curve for pure CR but is either negligible or of inverted am-
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plitude for the series of blends. In the mixtures and pure rubbers, the weight of this influence is largest in the pure CR sample and progresses nearly linearly through the mixtures to a value for the NBR sample of opposite sense. Figure 6B shows the near independence, in a chemical sense, of the second factor with respect to the third. The blends show a wide variation in the weighting of the third factor with only a small dependence on the second. The mixtures, on the other hand, show almost no dependence on the third factor. Figure 6C shows that the blends reflect similar influences of the first and third factors. A general pattern emerges that the TG
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curves for the blends are composed principally from factors one and three and those for the mixes from factors one and two. In order to enhance the interpretation of the extracted, VARIMAX rotation (24,25) was performed on the column cofactors to produce a wt of factors that might be more closely related to the TGs of the pure rubbers. Such a treatment is reasonable since TG curves for NBR and CR appear to be nearly orthogonal. The VARIMAX rotation maximizes the objective function,
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In these expressions, cij is ith of m loadings on the jth of p factors being rotated. The net effect is to maximize the sum of the squares of the individual loadings, thus emphasing the importance of a few large loadings at the expense of all others. The intention is to align the abstract factors with the more important "true" factors in the system, and in this case, such an alignment occurs. Figure 7 shows plots of the first 3 (of 10) row vectors that result from projecting the original row eigenvectors onto the coordinate system arising from the rotation of the original column eigenvectors. The second rotated vector shows a distinct similarity to the TG curve for pure CR. None of the rotated vectors aligns precisely with the NBR curve. This is likely to be a direct result of the character of the blends in the data set. Each blend contains apparently unaltered NBR as well as the NBR-CR adduct. This hinders the definitive assignment of behavior to one or the other component, and hence, discrete factors are not generated. The rotated column cofactors show a correspondingly interesting pattern (Figure 8). The blends and NBR show a clustering of nearly constant values (=0.07) on the factor 2 axes (Figure 8A,B). In contrast, the mixtures show a distinct change in the values of cofactor 2 with the CR content. Figure 8B most clearly illustrates the differences between the blends and mixes. Neither factor 2 nor factor 3 is strongly associated with NBR. This is evidenced by their small values associated with the point for NBR. The blends and mixes, however, show great divergence here. The blends are stronglyassociated with factor 3, and the mixes (and CR) with factor 2. These associations provide additional strong evidence that the CR originally present in the compounding mix is converted to an-
-1.0 d.6
.
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other form during some stage of the blending process. Figure 8C shows the mapping of the rubbers in the plane of rotated factors 1and 3. Importantly, there appears to be an inverse relationship (though apparently nonlinear) between the influence of factor 1, which we attribute in the main to NBR, and the influence of factor 3. This also supports the conclusion that the final, postblending character of the CR is indeed that of a adduct. Data Analysis via Constrained Variance Minimization. The results of the exploratory data analyses by SVD and VARIMAX rotation provide insight into the characteristic thermal behavior of the rubbers studied. These explorations lead to the question whether one can mathematically remove the influence of known components of the DTG curves while leaving the others intact. If possible, this would permit an estimation of the shape of the DTG curve for the "pure" NBR-CR adduct. Rank annihilation factor analysis (RAFA) (26) and partial least-squares (PLS) modeling (27) are two methods developed to attempt this type of estimation. Typically, RAFA is applied to property-property matrices. An example of this type of matrix is where chromatographic retention volume and wavelength are the row and column headings and each value in the matrix is an absorbance; that is, the entire matrix represents the response of a single sample. In the case of rubbers studied here, an object-property matrix is obtained; there is only a single vector associated with each sample; hence, RAFA is not applicable.
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 17, SEPTEMBER 1, 1991
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0.7781 0.4757 0.4123 0.3359
0.0952 0.1800 0.2214 0.3083
CVM estimate NBR CR 0.5309 0.2735 0.2783 0.2838
-0.0653 -0.0330 0.0248 0.0180
The PLS method requires only that responses for the components to be quantitated be available as part of a matrix of calibration vectors. Responses for the standards need not be available from pure samples, only present in the calibration set, preferably as a part of several samples. The application of PLS to the NBR-CR rubbers involved calculation of the PLS calibration matrix from the DTG signals for the pure and mechanically mixed TG samples. This matrix was then used to estimate concentrations for NBR and CR in the matrix of DTG vectors for the blended rubbers. These concentrations were then used as proportions of the corresponding pure rubbers to subtract from the matrix of DTGs for the blended rubbers. The result of the operation yielded a matrix of vectors that are representative of the DTG signals attributable to the NBR-CR adduct. Figure 9 shows plots for the PLS-generated DTG curves for the suggested adduct. A summary of the values found by the algorithm appears in Table IV. A disturbingly significant portion of each curve takes positive excursions (meaning an increase in sample mass) a t tempeatures (300 and 450 "C) corresponding c l w l y to the extrema in the curves for the pure rubbers. Apparently the PLS algorithm calculates concentrations of NBR and CR that are slightly in excess of the amounts actually present in the blended materials. The anomaly at 300 "C could be the result of slight misalignment of the curves in terms of temperature but the excursion at 450 "C is less easily explained. These anomalies led to an attempt to extract the adduct signal by using an alternative method, constrained variance minimization (CVM). The CVM method is based on several assumptions about the character of the data vectors composing the matrix. Firstly, removing the influence of any curve should reduce the variance of the points making up the vector, subject to the constraint that the "true" signal (by "true" is meant the signal in the absence of noise) should never become positive. Secondly, the real vector is composed of the true signal and some measure of noise that causes the baseline to fluctuate randomly about true zero. In estimating the amount of CR and
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Table IV. Concentrations of NBR and CR in the Blended Rubbers as Found by the Partial Least-Squares and Constrained Variance Minimization Procedures
,
CELSIUS TEMPERANRE
CELSJUS TEMPERANRE
Figwe 0. Proposed DTG curves for the NBF-CR adduct found by using PLS to subtract contributions from NBR and CR.
,
and CR.
NBR in the blended rubbers, these two assumptions me balanced by using a modified SIMPLEX algorithm (28). A vedor of initial estimates of the concentration ([CONC]) was chosen as a vector of the original amounts of appropriate rubber in the compounding mix before calendaring. The nonpositivity criterion was met by first calculating the rootmean-squared (RMS) value, C, of the positive points on each vector in the original data set. A residual matrix, [R], was constructed by subtracting amounts of pure rubber vector, [PI, from each of the original data vectors, as defiied by the current concentration test vector. In matrix notation, this calculation is [R] = [D] - [P][CONC] where [D] is the original data matrix. The penalty for positivity was constructed by calculating the RMS value (C') of the points in the residual matrix that were more positive than the corresponding points in the original data. Next, a correction factor (CF) was calculated as CF = C'/C Although several correction factor generation functions were tested in the minimization procedures, the one reported here seemed to give the most reasonable results. The objective function for the SIMPLEX minimization was defined as response = (1
+ CF)CVARi
where VARi is the variance of the ith column vector of the data matrix. SIMPLEX was then allowed to operate on the initial vector of estimated,concentrations, recalculating the response function for each estimate, until a minimum in the response function was found. The resulting test vector was then used as the concentration estimate for the appropriate pure rubber. The procedure was performed twice, once using NBR as the pure component and once using CR. The resulting concentration estimates are listed in Table IV along with those found using the PLS procedure. Once concentration estimates were obtained, the corresponding amounts of signal for the pure rubbers were subtracted from the data vectors representing the blends to estimate the shape of the DTG curve for the blend adduct. These estimates are plotted in Figure 10 and show similar results to the PLS-generated curves with the exception that the positive excursions in the curves are substantially reduced. Also of note is the smooth appearance of the CVM-generated curves in the region of the characteristic CR peak. The concentrations found for CR by CVM seem to be generally more reasonable than those found with PLS. Despite the absence of the characteristic CR signal in the original DTG
ANALYTICAL CHEMISTRY, VOL.
cruves for the blends, the PLS-based estimates show up to 30% CR, clearly not the case. Although there are small contributions, both positive and negative, in the concentration estimates by using CVM, they are reasonably close to zero, given the approximate nature of this procedure. The presence of two negative-going extrema in both the PLS and CVM results is interesting. This might be an indication that the decomposition of the NBR-CR adduct is a two-stage process. If one examines the ratio of height of the peak at 360 OC to that at about 440 “C for each resolved curve, there appears to be a regular decrease in ratio from about 8:2 to 4:6,on the same order as the change in ratio of CR to NBR in the initial materials themselves. This progression in ratio and inverse relationship between the heights of the two extrema may indicate that there are two adduct species present in the blends whose relative concentrations in the blended rubbers are dependent on the original ratio of starting materials.
CONCLUSION Factor analytical methods can clearly be coupled to “chemical intuition” to enhance the understanding of processes such as controlled pyrolysis. The real power of the use of factor-analytical methods in the analysis of complex chemical phenomena, such as the pyrolysis of rubber blends, lies in the ability to gain molecular chemical insights that might otherwise be obscured. The belief that temperature control and a knowledge of the process temperature is of major importance in a polymer blending process is confirmed in the analysis reported here. Further use of these modern mathematical methods could permit the prediction of such properties as flammability and could help discover why nominally identical materials from different batches are, in fact, very different in their performance in actual use. Studies are currently underway in our labs to test this potential. ACKNOWLEDGMENT The interest and encouragement of Robert Hannah is gratefully acknowledged. This work was carried out, in part, in a cooperativeresearch agreement between Duke University and the Institute of Chemistry, Academy of Sciences of the Republic of Estonia.
63,NO.
17, SEPTEMBER 1, 1991
1691
Registry No. ZnO, 1314-13-2; HSC(CH2)16COOH, 57-11-4; S, 7704-34-9; MgO, 1309-48-4; CBS, 95-33-0;diphenylguanidine, 90114-42-4.
LITERATURE CITED (1) Taylor, T. J.; Khanna, Y. P. Thennochim. Acta 1988, 136. 219. (2) Wendiandt, W. W. 7 l ” l Analysis, 3rd ed.,Chemical Analysis Series; Wiiey-Interscience: New York, 1988: Voi. 19, Chapter 2. (3) Larkin, D. E. I n Composihbnel Analysis by Thermcgravlmehy S7P 997; Earnest, C. M.. Ed.; American Society for Testing and Materials: Philadelphia, 1988; p 28. (4) Zeyen, Richard, L.. 111. Rubber WorM 1989, 199, 14. (5) Casa, F.; Glacometti Schieroni, A.; Massimino, P. Ind. G o m m 1988, 32, 22. (8) Guiochon, 0. Anal. Chem. 1961, 33, 1124. (7) Newkirk, A. E. Anal. Chem. 1960. 32. 1558. (8) Duval, C. Anal. Chlm. Acta 1964. 31, 301. (9) Simons, E. L.; Newkirk. A. E. Talenta 1984, 1 1 , 549. (10) Gibbons, J. J. Am. Lab. 1987, 19, 33. (11) Burns, D. H.; Callis, J. B.; Christian, G. D. Anal. Chem. 1986, 58, 2805. (12) TGA7 Thermogrevlmetric Analyzer, Operations Manual; Perkin-Elmer Corporation; Norwalk, CT, 1987; Chapter 9. (13) Garn, P. D.; Menis, 0.; Weiderman, H. G. J . Therm. Anal. 1981, 20, 185. (14) McGhie. A. R. Anal. Chem. 1989. 55, 987. (15) Savksky. A.; Golay, M. Anal. Chem. 1964. 36, 1827. (18) Steiner, J.; Termonia, Y.; Deltour, J. Anal. Chem. 1972, 44, 1908. (17) Newkirk, A. E.; Simons, E. L. Talenta 1966, 13, 1401. (18) Golub, G.; Van Loan, C. F. Matrix Computations;Johns Hopkins University Press: BaItImore, MD, 1989; Chapters 2 and 8. (19) Press, W. H.; Flannery, B. P.; Teukolsky, S. A,; Vetteriing, W. T. Nu&cal Recipes h C ; Cambridge University Press: Cambridge, 0. B., 1988; Chapters 2 and 14. (20) Lawson. C. L.; Hanson, R. J. Solving Least Squares problems; PrentiaHali Series in Automatic Computation, Prentice-Hall: Engiewood Cliffs, NJ, 1974; Chapters 3-8. (21) Malinowski, E. R.; Howery, D. G. Factor Analysis in Chemisby; John Wlley 8 Sons: New York, 1980; p 82. (22) Malinowski, E. R. J . C h e m m . 1987, 1 , 33. (23) Wold, Svante. Technometrics 1987, 20, 397. (24) Kaiser, H. F. Psychometrlko 1957, 23, 187. (25) Afifi, A. A.; Azen, S. P. Stallstlcal Analysis-A Computer OnLMted Approach; 2nd ed.; Academic Press: New York. 1979; p 334. (28) Ho, C.-N.; Christian, G. D.; Davidson. E. R. Anal. Chem. 1978, 50, 1108. (27) Wold, S.; Martens, H.; Wold, H. I n Matrix pencils: proceedings o f a conference hekl at Pite Havsbad, Sweden; Lecture Notes in Mathematics No. 973; Kdagstrom, B., Ruhe. A,, Eds.; Springer-Verlag: New York, 1983, p 288. (28) Neider, J. A.; Mead, R. Compuf. J . 1965, 7 , 308.
RECEIVED for review March 23, 1990. Revised manuscript received April 26,1991. Accepted June 6,1991.
