Anal. Chem. 1980, 52, 311-314
(20) Gundermann, K-D. Angew. Chem. 1965, 7 7 , 572-580. (21) White, Emil H.; Zafiriou, Oliver; Kagi, Heinz H.; Hill, John H. M. J . Am. Chem. SOC.1964, 86,940-41. (22) Babko, A. K.; Terietskaya, A,; Dubovenko, L. I . Ukr. Khim. Zh.1966, 32, 728-31. (23) Kachibaya, V. N.; Siamashvili, I. L.; Mamukashvili, M. V. Ukr. Khim. Zh. 1971, 26, 1846-49. (24) Babko, A. K.; Markova, L. V.;Lukovskaya, N. M. J. Anal. Chem. USSR 1968, 23, 330-35. (25) Seitz, Rudolf W.; Hercules, David M. Anal. Chem. 1972, 44, 2143-49. (26) Babko, A. K.; Lukovskaya, N. M. Ukr. Khim. Zh. 1962, 17, 50-52. (27) Seitz, Rudolf W.; Suydam. Wallace W.; Hercules, David M. Anal. Chem. 1972, 4 4 , 957-63. (28) Anderson, Howard H.; Moyer, Rudolph H.; Sihbett, Donald J.; Sutherhnd, David C. U S . Patent No. 3659100,Aug. 14, 1970.
311
(29) Anderson, Howard H.; Moyer, Rudolph H.; Sihbett, Donald J.; Sutherhnd, David C. U S . Patent No. 3700896,Oct. 24, 1972. (30) Nederbragl, G. W.; Van der Horst, A,; Van Duijn, J. Nature (London) 1965, 206, 87. (31)Pitts, James N., Jr.; Fuhr, Hartmut; Gaffney, Jeffrey; Peters, John W. hviron. Sci. Techno/. 1973, 7 , 550-52. (32) Regener, V. H. J . Geophys. Res. 1964, 69,3795-3800. (33) Burdo, Timothy G.;Seitz, W. Rudolf Anal. Chem. 1975, 4 7 , 1639-43. (34)White, Emil H.; Bursey, Maurice M. J . Am. Chem. Soc. 1964, 86, 941-42.
RECEIVED for review June 21, 1979. Accepted November 6, 1979.
Factor Analysis of Some Physical and Structural Properties Influencing the Fluorescence Lifetimes of an Atabrine Homologous Series L. J. Cline Love,* Patricia Cala Tway, and Linda M. Upton Department of Chemistry, Seton Hall University, South Orange, New Jersey 07079
The technique of factor analysis has been applied to a matrix of fluorescence lifetime data as a function of solvents and solutes. The data factor analyzes as log T with two factors. Two properties of the solute, identity and number of carbon atoms attached to the exocyclic nitrogen, were found to reproduce the experimental data within experimental error. Many solvent properties were also studied, but no combination could be found whlch successfully reproduced the data. New insights into possible solvent effects were obtained however.
The fluorescence lifetime, r , of a compound in a solvent is dependent on many factors. This is evident from the definition of T , as given in Equation 1,
1 T =
kf + k i + kz +
(1) kq[Ql
where kf is the rate constant for fluorescence, k l is the rate constant for internal conversion, k, is the rate constant for intersystem crossing, and kq is the rate constant for any quenching reaction by the quencher Q. The value of each of these constants is, itself, dependent on a number of factors, although the exact functionality is seldom known. I t is apparent that in the absence of all decay processes competing with kf, the fluorescence lifetime is a maximum value. As more mechanisms are made available for nonfluorescence deactivation of the molecule, the measured fluorescence lifetime becomes shorter. T h e fluorescence lifetimes of compounds can be very sensitive indicators of solute characteristics such as structural conformation and of solvent properties. When experiments are performed to study some of the effects on lifetimes of a particular set of compounds in certain environments, attempts are made to hold most of the factors constant, while one or two of them are studied (1-4). Because of the large number of possible factors, some often unknown, this is a difficult task to perform reproducibly and with any degree of certainty. For example, one may hold temperature, purity of environment, excitation wavelength, and compound constant in the hopes 0003-2700/60/0352-031 l$Ol .OO/O
of studying solvent effects. However, many factors do change when a solvent is changed, for example, viscosity, pH, ionization of the sample, solvent polarity, hydrogen bonding, ionic strength, and quenching. Additional information on solvent effects is obtained by studying a homologous series of compounds in different solvents. One can quickly build a matrix of fluorescence lifetimes, with solvents as the columns and compounds as the rows. It is often difficult to understand and explain the trends seen in such a data set because several different factors can contribute to the data simultaneously. Factor analysis is a mathematical technique which enables the chemist t o see and explain trends in a data set multidimensionally ( 5 ) . I t is, in part, a multiparameter curve-fitting method (6). It has been used in chromatography (7, 8 ) , spectroscopy ( 9 ) ,mass spectrometry ( I O - I Z ) , and biological activity (13). Factor analysis allows one to (1)determine the number of abstract factors needed to reproduce the array of data, (2) correlate these abstract factors with actual physical properties of the row and column designees (target transformation), and (3) predict new data using these target transforms. It has been difficult to utilize factor analysis in the past because of the uncertainity in how and when to use it ( 1 4 , 1 5 ) . The error theory of Malinowski (16-18) has solved some of these problems. This theory provides semiquantitative indicator functions which can be used to evaluate results of factor analysis. T h e use of Malinowski’s indicators lends assurance to the chemist that the problem under consideration is indeed factor analyzable and that the correct number of factors are chosen for the problem. I t was felt that a matrix of fluorescence lifetimes with solvents and solutes as the row and column designees might be amenable to factor analysis. The criteria to use in determining whether a problem is factor analyzable are (1)the components, K , of the matrix should be related to energy, (2) each component should be related to a linear sum of terms which are products of factors r and c, as given below,
Kl., = xrl.,c)h J=1
(2)
where the r factors relate solely to the row designees and the 0 1980 American Chemical Society
312
ANALYTICAL CHEMISTRY, VOL. 52, NO. 2, FEBRUARY 1980
Table I. Data Matrix of Fluorescence Lifetimes (ns) of a Homologous Series of Atabrine in Several Solvents solvents CHC1,/ CH,OH/ acetic acid acetic acid heptane 0.1 N HCl CHC1, C,H,OAc compounds CH,OH ~
18.7 10.4 7.2 7.9 6.9 8.7 7.5 7.5 7.6
1
2 3 4 5
6 7 8
9
13.0 8.3 10.4 6.6 4.9 6.8 5.8
5.9 5.9
11.2 8.6 6.5 4.6 4.3 4.5 4.6 4.3 4.2
20.1 12.7 6.7 3.6 3.2 2.9 2.3 1.6
5.9
4.4 4.1 2.1 1.9 2.0 2.0 1.9 2.1
1.3
19.9 15.0 7.4 4.0 3.8 3.3 2.7 2.1 2.7
14.0 10.7 12.7 12.9 13.1
13.4 14.3 13.5 14.2
Table 11. Results of t h e Covariance Factor Analysis of the Logarithm of the Fluorescence Lifetimes
I
I1
Figure 1. Structures of atabrine homologues. In the tables, compound Icorresponds to structure I, compound 5 is atabrine where n = 3 and R = CH, in structure 11, and compounds 2-4, 6-9 correspond to n = 1-3, 4-7, and R = H in structure I1
c factors only t o the column designees and, finally, (3) the dimensions of the matrix should be large (19). Since a fluorescence lifetime is an inverse rate constant, it was thought that In T should be related to energy (20). It was also felt that the rate constant dependence on the solutes should be independent of its dependence on solvents, and that any effects should be additive. This paper presents the results of the application of factor analysis t o a matrix of fluorescence lifetimes of a homologous series of atabrines in a series of solvents, taken from the data in reference 4. T h e results obtained support previous interpretations ( 4 ) and give some new insight into the effects of environment on the fluorescence lifetimes of these molecules.
EXPERIMENTAL The method of collection of the fluorescence lifetime data and the conditions used for the atabrine series have been described previously ( 4 ) . The structures of the atabrine homologues studied are shown in Figure 1. The error in T of *0.3 ns produces a median error of f0.05 ns in In T. The factor analysis (FA) computations were carried out on an IBM 360 computer at Merck & Co., Rahway, N.J. The computer program was obtained from D. G. Howery of Brooklyn College. The mathematics and basic assumptions of FA have been detailed elsewhere (21,22). Covariance FA was used throughout this work because the errors in the In T data are for the most part constant (22).
The number of factors which span the space was determined using the error theory of Malinowski (16,17). In this theory three different functions are calculated which can be used to evaluate the number of factors spanning the data set. The real error (RE) is the difference between the pure data and the experimental data set, and should equal the experimental error of the measurements. Theoretically the imbedded error (IE) should reach a minimum plateau when the correct number of factors is used. Additional factors are merely redundant and do not remove any further error. The indicator function (IND) has been found empirically to reach a minimum when the correct number of factors are employed. Both IND and IE were monitored in this work as each additional factor was employed. When the IE began to decrease more slowly than previously and the IND reached a minimum, the number of factors was taken. Target transforms were then performed using target vectors of various physical or chemical properties of the solutes and the solvents. The validity of these transformations was evaluated by comparison of the target vector with the predicted vector (22) and by use of Malinowski’sSPOIL and RELI (reliability) functions
abstract factor
real error
1
0.1742 0.0493 0.0400 0.0311 0.0187
2 3 4 5
extracted imbedded indicator error error (IE) error (IND) 0.1613 0.0417 0.0303 0.0204 0.0100
0.0659 0.0264 0.0262 0.0235 0.0158
0.0048 0.0020 0.0025 0.0035 0.0047
(18,23). The RELI function is used as a measure of how close the target vector matches a true factor. The SPOIL function measures the difference in the error in the reproduced data matrix if an abstract eigenvector is replaced with the target vector. A target vector having a SPOIL between 0 and 3, and/or a RELI value between 0.5 and 1 was considered to be a target vector that reproduced as an abstract eigenvector. Combinations of the various vectors which seemed to be factors from the target transformations, were used to try to reproduce the data matrix. This process indicates if the successful target vectors are independent of one another, and hence if they can be used to theoretically determine new data points for the fluorescence lifetime data matrix. The combination was deemed successful when the data matrix was reproduced with an RMS error approximating the experimental error and the Exner function (14) was less than 0.5.
RESULTS AND DISCUSSION Several forms of the data were factor analyzed including T , log T , T - ~ /and ~ , 7-lI2. T h e only form which was found to be factor analyzable, Le., gave a reasonable number of factors with the proper behavior of IE and IND, was the log T as expected. The data matrix of T is presented in Table I. The I E and IND data are shown in Table 11. One can see t h a t each error function clearly indicates that two abstract factors explain the data. The R E (real error) given at the correct number of factors for the data set is the real error predicted for the data (16,17). It is seen in Table I1 that the predicted R E in log T is f0.05 which corresponds t o a n error of f0.12 in In 7. The estimated experimental error in In T (f0.05) agrees fairly well with the R E predicted. The conclusion reached in this first FA step is that two abstract factors explain the In fluorescence lifetime data, i.e., two properties of the solutes and two properties of the solvents. The eigenvalues found for these abstract factors were 41 and 1.5, indicating that one factor is thirty times more significant than the other. In an attempt to correlate these abstract factors with physically significant properties of the solutes and solvents, some target transformations of various solute and solvent properties were performed. The results of the transforms for some solute properties are shown in Table 111. T h e transformation is evaluated by a comparison of the test vector with the predicted vector and by use of the SPOIL and RELI functions. As can be seen in the Table, the number of carbon atoms (attached t o the exocyclic nitrogen group), solubility
ANALYTICAL CHEMISTRY, VOL. 52, NO. 2, FEBRUARY 1980
313
Table 111. Target Transformation of Solute Properties no. of carbon atomsa compound test predicted
a
1
0
2 3 4 5 6 7 8 9 SPOIL RELI
2 3 4 5 5 6 7
0.9 0.7 2.8 4.8 4.8 5.6 6.1 7.0 7.1
8
solubility test predicted 3 4 5 5 6 4 3
2 1
1.5 0.5
17.8
PKC test predicted -
4.8 4.1 3.9 3.4 3.2 3.4 3.3 3.2 3.2
7.29 7.78 7.86 7.69 8.06 8.17 8.21 8.26
0
-
polarityC test predicted
8.63 7.37 7.80 7.81 7.49 8.14 8.14 8.26 8.40
3.1 2.6 5.9
0
5.9 7.3 8.4 7 .o
8.8
8.7 10.1 10.9 12.2 12.5
9.8
11.0 12.3 13.5
The number of carbon atoms attached t o the exocyclic nitrogen group.
identity test predicted
____-____
1.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.9 1.0 1.0 0.9 1.0 1.0 1.o 1.0
2.7
1.2
0.1
0.7
Solubility in water ( 2 4 ) .
Reference 25.
Table IV. Target Transformation of Solvent Properties
solvent CH,OH CHC1, C, H,OAc heptane 0.1 N HCl HOAc/CH,OH HOAclCHCl, SPOIL RELI
Eta (DimrothReichardt ) test predicted
56 35 38 31 76 56 39
45 43 43 36 66 63 38 3.7 0.1
(overall strength) test predicted 6
12.9
9.0
0.60 0.57 0.45 0.32
17.0 16.0 8.4
0.65 0.55
10.7 10.4 10.5
5.1
8.6 7.4 21.0 12.9 9.2
viscosityC test predicted 0.58 0.55 0.54 0.44 0.81 0.78 0.50
1.00
4.4 0.2
test
PHd predicted 0.10 0.02 0.14 0.38 0.99 0.78 0.57
0 0 0 0 1 1 -
4.5 0.3
nD
test
predicted
1.33 1.44 1.37 1.39 1.34 1.33 1.44
1.45 1.36 1.26 0.88 1.47 1.50 1.50
3.6 0.4
6.4 0.3
a Dimroth-Reichardt polarity function ( 2 6 ) . Polarity expressed as overall strength ( 2 8 ) . Viscosity ( 2 9 ) . when there are no available free protons; pH = 1 when there are available protons. e Refractive index ( 2 9 ) .
in water ( 2 4 ) ,p K (25),polarity (25),and identity were tried as possible solute factors. The values for these physical properties were obtained from the referenced works. The number of carbon atoms and identity transformed the best as one of two factors with spoils of 1.5 and 1.2, respectively. Polarity also worked relatively well with a SPOIL of 2.7. RELI values were, respectively, 0.5, 0.7, and 0.1. No similar success was obtained in testing solvent properties. The results of the best solvent vector transformations are given in Table IV. Expressions of solvent polarity in the forms of Et (Dimroth-Reichardt numbers) (26),e (dielectric constant) (27),and b (overall strength) (28),were tried as well as viscosity (29), hydrogen bonding ability (28), hydrogen donating ability (28),and refractive index, nD (29). The pH expressed as 0 when no free protons were available and 1when they are available was also used. The p H value for acetic acid-chloroform was free-floated. No definitive results were obtained from these data. All SPOIL functions exceeded 3.0 and all RELI values were less than 0.5. Based on Malinowski's interpretation of the SPOIL and RELI functions, Et and p H were fair transforms and the other vectors were poor (18). T o ascertain whether the physical factors which correlated with the abstract factors were independent of one another, attempts were made t o reproduce the data matrix within experimental error using combinations of these factors one, two, and three at a time. Some of the results of the solute vector combinations are given in Table V. Keeping in mind that the R E is 0.05, it may be seen t h a t the identity vector when used as the only factor, does a fairly good job of reproducing the data with RMS equal to 0.21 and an Exner value of 0.74. However, identity plus the number of carbon atoms reproduces the data within experimental error; RMS is equal to 0.08. These results seem to indicate that identity is the primary factor associated with an eigenvalue of 41 and
pH = 0
Table V. Combinations of Solute Factors pro pert ies one factor two factors three factors
a
no. of carbon atoms polarity identity polarityiidentity no. of C atoms/identity pKiidentity no. of C atomsiidentityl polarity no. of C atoms/identity/ PK
RMS ExneP 0.65 0.56 0.21 0.10 0.17 0.11
2.19 1.92 0.74 0.35 0.26 0.69 0.39
0.64
2.47
0.08
Reference 14.
the number of carbon atoms is the secondary factor with a n eigenvalue of 1.5. Adding a third factor worsens t h e reproduction of the data with a RMS of 0.11 and an Exner of 0.39. The results using the polarity vector are only slightly worse than those using the number of carbon atoms vector. In addition, using three factors consisting of carbon atoms, identity, and polarity reproduces the data fairly well. This would indicate that the polarity of these atabrine homologues is dependent on the number of carbon atoms attached to the exocyclic amino group. These results seem t o corroborate the previous interpretation of these data ( 4 ) . In the majority of solvents used, most of the atabrine homologues behave similarly and have approximately the same fluorescence lifetimes (Table I). T h e only molecular difference in the series--the number of carbon atoms attached to the exocyclic nitrogen-does not affect any radiative or nonradiative transitions of the excited state. The same fluorophore is common to each member of the series. However, in two of the solvents the molecular form is protonated. I t has been suggested ( 4 ) that in these solvents
314
ANALYTICAL CHEMISTRY, VOL. 52, NO. 2, FEBRUARY 1980
~
Table VI. Combinations of Solvent Factors one factor
properties
RMS
Exnera 5.17 0.83
viscosity pH/viscosity pHinD
1.55 0.25 0.33 0.24 0.24 0.27 1.53 0.24
0.85
PH nD
two factors
P y Et
three factors
viscosity/Et viscosity/nD/pH
1.18 0.81
0.84 0.90 5.11
Reference 14. the fluorophore is changed through protonation to include the increasing carbon chain. This chain, then, affects the fluorescence lifetime through increasing internal conversion. This interpretation would indicate that to produce these data all of the members of the series behave similarly except in acidic hydrogen bonding solvents where carbon chain length becomes important. In other words, identity among the solutes is the primary factor, while carbon atom number is secondary. T h e best results for the solvent vector combinations are given in Table VI, The experimental data could not be reproduced within experimental error using one, two, or three target vectors. The best result was a RMS of 0.24. The pH vector seems to be a necessary factor, although not the primary one. It reproduces the data the worst of all vectors when used alone (Exner = 5.17), yet only when it is used in combination with another vector does the Exner value fall below 1.0. This again supports the previous interpretations of this data in that the presence of free protons (pH 1)affects the molecule in such a way as to alter the fluorescence transition ( 4 ) . Therefore, this p H vector should be necessary to explain a minor part of the lifetime data. Using two factors, the best vector combinations are pH and viscosity, and p H and refractive index. If p H is accepted as a minor factor, the trend in the data which must be explained by the major factor is the change in T with solvent across Table I from left to right. Viscosity is known to affect fluorescence lifetimes both through polarization effects and through effects on internal conversion (30). The refractive index of a solvent , f0 l / n 2 (31). is related to the natural lifetime, T ~through It is possible that either viscosity or refractive index is a solvent factor in these lifetime data, but that the functional form of the vector is incorrect. For example, for some dye molecules it has been found that the fluorescence lifetimes are proportional to the 2 / 3 power of the viscosity of the solvent (32). However, based on the available data, no definitive conclusions about the solvent physical factors associated with the two abstract factors predicted by factor analysis can be reached.
spectroscopic data on large molecules with a common chromophore will generally have the identity vector as a target vector because of this commonality. Finally this study has shown that analytical chemists with a rather modest working knowledge of factor analysis and computer programming can easily make profitable use of this technique to aid in interpretation of complex, multivariable data.
ACKNOWLEDGMENT The authors thank D. G. Howery for supplying the factor analysis computer program, H. B. Woodruff for discussions and assistance in using the factor analysis program, and the Automation and Control Department of Merck and Co., Rahway, N.J., for assistance and use of their computer facilities.
LITERATURE CITED G. W. Robinson, G. R . Fleming, A. W. E. Knight, J. M. Morris, and R. J. S. Morrison, J . Am. Chem. SOC., 99, 4306 (1977). A. Bowd, J. B. Hudson, and J. H. Turnbull, J. Chem. Soc., Perkin Trans. 2 , 1973, 1312. S. Babiak and A. C. Testa, J . Phys. Chem., 80, 1882 (1976). L. J. Cline Love, L. M. Upton, and A. W. Ritter 111, Anal. Chem., 50, 2059 11978). D. G.Howery, Am. Lab., 8(2), 14 (1976). P. H. Weiner, Chemfech., 321 (1977). P. H. Weiner and D. G. Howery, Can. J . Chem., 50, 448 (1972). R. B. Selzer and D. G. Howery, J . Chromatogr., 115, 139 (1975). 2 . 2.Hugus and A. El-Awady, J . Phys. Chem., 75, 2954 (1971). G. L. Ritter, S. R. Lowry, T. L. Isenhour, and C. L. Wilkins. Anal. Chem., 48, 591 (1976). R. W. Rozett and E. M. Petersen, Anal. Chem., 47, 1301 (1975). F. J. Knorr and J. H. Futrell, Anal. Chem., 51. 1236 (1979). M. L. Weiner and P. H. Weiner, J . Med. Chem., 18, 655 (1973). J. H. Kindsvater, P. H. Weiner. and T. J. Klingen, Anal. Chem., 46, 982 (1974). P. H. Weiner, H. L. Lao, and 8. L. Karger, Anal. Chem., 48, 2182 (1974). E. R. Malinowski, Anal. Chem., 49, 606 (1977). E. R. Malinowski, Anal. Chem., 49, 612 (1977). E. R. Malinowski, Anal. Chim. Acta, 103, 339 (1978). D. G. Howery, Brooklyn College, Brooklyn, N.Y., personal communication, 1976. A. A. Frost and R. G. Pearson, "Kinetics and Mechanisms", John Wiley, New York, 1961, p 89. P. H. Weiner, E. R. Malinowski, and A. R. Levinstone, J . Phys. Chem., 74, 4537 (1970). D. G. Howery, in "Statistics", R. F. Hirsch, Ed., Franklin Institute Press, Philadebhia. Pa.. 1978. ChaDter 7. E. d.hrl&linowski: Steve'ns Inbtitute of Technology, Hoboken, N.J., personal communication, 1979. G. V. Downing, Merck 8 Co., Rahway, N.J., personal communication, 1977. J. L. Irvin and E. M. Irvin, J . Am. Chem. SOC., 72, 2743 (1950). A. Gordon and R. Ford, "The Chemist Companion", John Wiley 8 Sons, New York, 1972. D. T. Sawyer and J. L. Roberts, "Experimental Electrochemistry for Chemists", John Wiley 8 Sons, New York, 1974. L. R. Snyder and J. J. Kirkland. "Modern Liquid Chromatography", John Wiley 8. Sons, New York, 1974. R. C. Weast, Ed., "Handbook of Chemistry and Physics". Chemical Rubber Co., Cleveland, Ohio, 1970. G. G. Guilbault, "Practical Fluorescence", Marcel Dekker, New York, 1973. S. J. Strickler and R. A. Berg, J . Chem. Phys., 37, 814 (1962). W. Yu, F. Pellegrino, M. Grant, and R. R . Alfano, J . Chem. Phys., 67, 1766 (1977).
CONCLUSIONS This paper presents a new application of factor analysis to fluorescence decay rate data. Although some solute factors were successful in reproducing the data matrix, the solvent properties commonly used for spectroscopic interpretations were not found to be factors. The results of factor analysis suggest a possible effect of viscosity on these lifetime data that has not been previously considered. Also, factor analysis of
RECEIVED for review July 30, 1979. Accepted November 8, 1979. The financial assistance of the Analytical Division of the American Chemical Society through an award of a 1976-77 full year fellowship to L.M.U. is gratefully acknowledged. Research support provided by the State of New Jersey under provisions of the Independent Colleges and Universities Utilization Act is also gratefully acknowledged.
