146
Anal. Chem. 1991, 63, 146-15 1
It is suggested that the critical values of Q and the related r criteria which have been generated for the 95% confidence level should be used routinely by practicing analytical chemists in testing for the rejection of outliers since this confidence level provides a reasonable compromise between ultraconservatism and the overzealous rejection of deviant values. These values should also be incorporated into future analytical chemistry treatises and textbooks dealing with tests for the rejection of data to provide a uniform set of critical values a t this standard confidence level. As a concluding comment, it should be noted that recent studies on the variance of the arithmetic mean after rejection of outliers suggest the superiority of two more recently proposed criteria (Huber-type skipped mean and Shapiro-Wilk rules) for rejection decisions (19),particularly for larger samples containing multiple outliers. Nonetheless, the simplicity of Dixon's range ratio tests argues strongly for their continued use in many analytical applications.
ACKNOWLEDGMENT I express my appreciation to my colleagues, George Schenk and David Coleman, for helpful suggestions regarding this manuscript. LITERATURE CITED (a) Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry. 2nd ed.; Wiley: New York. 1988; pp 62-64, 218. (b) Anderson, R. L. Practical Statistics for Analytical Chemlsts; Van Nostrand Reinhoid: New York, 1987; pp 31-32. (c) Caulcutt, R.;Boddy, R . Statistics for Analytical Chemists; Chapman and Hall: London, 1983; pp 66-67, 248. Barnett, V.; Lewis, T. Outliers in Statistical Data, 2nd ed.; Wiiey: New York. 1984. (a) Skoog, D. A.; West, D. M.; Holler, F. J. Analytical Chemistry: An Introduction, 5th ed.; Saunders: Philadelphia, 1990; p 56. (b) Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, 5th ed.; Saunders: New York, 1988; pp 13-16. (c) Hargis, L. G. Analytical Chemistry; Prentice-Hall: Englewocd Cliffs, NJ, 1988; p 56. (d) Fritz, J. S.:Schenk, G. H. Quantitative Analytical Chemistry, 5th ed.: Allyn 8 Bacon: Boston, 1987; pp 45-46. (e) Potts, L. W. Quan-
titative Analysis: Theory and Practice; Harper 8 Row: New York, 1987; pp 78-80. (f) Rubinson, K. A. Chemical Analysis; Little, Brown: Boston, 1987; pp 162-164. (9) Day, R. A., Jr.; Underwood, A. L. Quantitative Analysis, 5th ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986; pp 29-31. (h) Manahan, S.E. Quantitative Chemical Analysis; Brooks Cole: Monterey. CA, 1986; pp 74-75. (i)Kennedy, J. H. Anaiyticai Chemistry: Principles, 2nd ed.; Harcourt, Brace, Jovanovich: New York, 1990; pp 35-39. (i) Harris, D. C. Quantitative Chemical Analysis; Freeman: San Francisco, 1982; pp 51-52. (k) Ramette, R. W. Chemical Equilibrium and Analysis ; Addison-Wesley: Reading, MA, 1981; pp 53-54. (I) Christian, G. D. AnalyticalChemistry; Wiley: New York, 1980; pp 78-79. (m) Flaschka, H. A,; Barnard. A. J., Jr.; Sturrock, P. E. Quantitative Analytical Chemistry; Willard Grant: Boston, 1980; pp 19-20. Dixon, W. J. Ann. Math. Stat. 1950, 27. 488-506 King, E. P. J. Am. Stat. Assoc. 1953, 48, 531-533. Deming, W. E. Statistical Aaustment of Data; Wiley: New York, 1943 (republished by Dover: New York. 1964); p 171. Parratt, L. G. Probability and Experimental Errors in Science ; Wiley: New York, 1961; pp 176-178. Hawkins, D. M. Identification of Outliers; Chapman and Hall: London, 1980. Beckman, R . J.; Cook, R . D. Technometrics 1983, 25, 119-149. (a) Burr, I. W. Applied Statistical Methods; Academic: New York. 1974; pp 194-195. (b) Hamilton, W. C. Statistics in Physical Science; Ronald: New York. 1964; pp 45-49. Dixon, W. J. I n Contributions to Order Statistics; Sarhan, A. E., Greenberg, B. G., Eds.; Wiley: New York, 1962; pp 299-342 (see pp 314-317). Natrelia, M. G. Experimental Statistics ; National Bureau of Standards Handbook 91; NBS: Washington, DC, 1963; Chapter 17. "Student". [Gossett, W. S.] Biometrika 1908, 6 , 1. Taylor, J. R. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements; University Science: Mill Valley, CA, 1982; pp 142-145. Dixon, W. J. Ann. Math. Stat. 1951, 22, 68-78. (a) Dixon, W. J. Biometrics 1953, 9, 74-89. (b) Dean, R . 8.; Dixon. W. J. Anal. Chem. 1951, 2 3 , 636-638. Peters, D. G.; Hayes, J. M.; Hieftje, G. M. Chemical Separations and Measurements : Theory and Practice of Analytical Chemistry; Saunders: Philadelphia, 1974; p 36. Youmans, H. L. Statistics for Chemistry; Charles E. Merrill: Columbus, OH, 1973; p 65 ff. Hampel, F. R. Technomefrics 1985, 27, 95-107.
RECEIVED for review June 15, 1990. Accepted October 16, 1990.
Frequency-Domain Spectroscopic Study of the Effect of n-Propanol on the Internal Viscosity of Sodium Dodecyl Sulfate Micelles M. J. Wirth,* S.-H. Chou, and D. A. Piasecki Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware 19716 The rotational diffusion behavior of tetracene in sodium dodecyi sulfate micelles is studied as a function of the npropanol content of the micellar solution. Fluorescence anisotropy measurements using frequencydomain spectroscopy show that tetracene reorients faster with increasing concentration of n-propanol. This result Is consistent with micellar liquid chromatographic studies and lends insight into the role of n-propanol as a mobile-phase modifier. The components of the rotational diffusion tensor are determined from the double-exponential anisotropy decay. These confirm the validity of the Debye-Stokes-Einstein model and allow caicuiation of the viscosity of the micelle interior. The viscosity decreases from 8 to 4 CP as the concentration of n-propanol increases from 0 to 10% (v/v). The components of the rotational diffusion tensor indicate that the solvation environment of tetracene in the micelle is structurally disordered.
* T o whom correspondence should be addressed. 0003-2700/91/0363-0146$02.50/0
INTRODUCTION The importance of micelles in analytical chemistry has burgeoned both in spectroscopy and in separation science. Micelles are used to enhance fluorescence ( I ) , thermal lensing (21, and room-temperature phosphorescence ( 3 ) . In liquid chromatography, micelles modify the organic content of the mobile phase ( 4 ) , allowing more rapid gradient elution ( 5 ) . Micelles can serve as the pseudostationary phase in electrokinetic chromatography (6), providing selectivity for separation of polar organic solutes. The nature of solute interactions with micelles is thus a vital area of research. An important question entails the dynamics of solutes interacting with micelles. One of the drawbacks of micellar liquid chromatography has been poor column efficiency due to slow mass transfer between the surfactant-modified stationary phase and the mobile phase ( 7 ) . Dorsey et al. demonstrated that the addition of at least 1% of n-propanol gives a 40% increase in the number of theoretical plates for the solute benzene in micellar liquid chromatography (8). They 1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
attributed the effect to better wetting of the hydrophobic phase. Similar results were obtained by Yarmchuk et al., who interpreted the effect as an increase in the residence time of the hydrophobic solute in the interstitial solution (9). Borgerding et al. found an improvement of roughly a factor of 2 higher and related the effect to a decrease in the amount of surfactant adsorbed onto the surface (10). Clearly, npropanol improves chromatographic efficiency, but a better understanding of the fundamental physical nature of mass transfer in micellar media is needed. As a step toward the fundamental understanding of solute dynamics in micellar media, the effect of n-propanol on the viscosity of the interior of sodium dodecyl sulfate (SDS) micelles is studied spectroscopically. The numerous spectroscopic studies of micelle properties have been reviewed recently (11). Applications of various techniques to sense viscosity have shown considerable disagreement, with 4 , 5 , 9 , 10, 11.5, and 19 CPbeing reported for SDS (12-17). Determination of viscosity in microscopic media is thus a challenging problem in spectroscopy. Rotational diffusion behavior provides more information about viscous media than does translational diffusion behavior. Rotational diffusion studies provide three degrees of freedom, D,, Dy, and D,, which are the rotational diffusion coefficients for the three solute axes. Translational diffusion measurements provide only one degree of freedom, D,, which is the translational diffusion coefficient. The additional information is necessary for interpreting viscosity from D. For systems that are described by the Debye-Stokes-Einstein (DSE) model, one cannot calculate q from D, without knowing whether the stick or slip boundary condition applies (18). Rotational diffusion studies reveal whether the slip or stick and boundary condition applies: the relative values of D,, Dy, D, are very different for the stick and slip boundary conditions. A more urgent advantage applies to studies of microscopic media: the relations between Dt and 7 are valid only for media that behave according to the DSE model, i.e., viscous, continuum fluids. The measurement of D, alone carries no internal check on the applicability of the DSE model to new environments. Rotational diffusion studies can assess the applicability of the DSE model through the relative values of D,, D,, and D,. The internal check intrinsic to rotational diffusion makes it particularly valuable to microscopic media. Rotational diffusion is typically sensed by using fluorescence depolarization measurements. Because SDS is so fluid, steady-state fluorescence depolarization measurements are not applicable; solutes reorient very fast compared to their fluorescence lifetimes. Ultrafast techniques lack the high sensitivity needed to avoid comicellization. The requirements of high sensitivity and picosecond time resolution have been satisfied by the development of frequency-domain fluorometry for fluorescence anisotropy measurements (19, 20). This method has been applied to the study of the reorientation of a cationic organic solute attached to an SDS micelle (21) but has not yet been applied to the study of the rotational diffusion of nonpolar solutes inside SDS micelles. The purpose of this work is to study the influence of small amounts of n-propanol on the rotational diffusion of tetracene in an SDS micelle, to reveal any changes in the viscosity of the interior of the micelle. The rotational diffusion behavior is studied in detail to gain insight into the process of diffusion inside the micelle. Frequency-domain fluorescence depolarization measurements capable of picosecond time resolution are employed. Tetracene is chosen because it is very hydrophobic, eliminating significant contribution from the aqueous medium. The rotational diffusion behavior of tetracene is studied for SDS solutions above the critical micelle concentration as a function of added n-propanol.
147
THEORY In fluorescence depolarization measurements the intensity of fluorescence is detected as a function of time and polarization. In isotropic media, such as micellar solutions, the fluorescence anisotropy, r ( t ) ,is defined from the intensities of light emitted with polarization parallel (Ill) and perpendicular (I,) to the excitation polarization.
Substitution of the cos20 relation for absorption of polarized light shows that r ( t ) is the correlation function of the angle between the excitation and emission transition moments. (P2(cos 0) is the second Legendre polynomial.)
r ( t ) = 0.4(P2(cos 0 ) )
(2)
The accepted model for describing the angular correlation functions of medium-sized organic solutes is the DSE rotational diffusion model, where the solute angle changes by very small increments due to the high collision rate with the surrounding solvent molecules (22). The rotational diffusion coefficient, D , is related to the decay constant of the angular correlation function, rO,,by ror= 1/6D. The relation of these parameters to the viscosity is
TV
?or
=k d 8 t i c k C
(3)
where V is the hydrodynamic volume of the solute and kT is the thermal energy. The factor istick depends upon solute geometry, as derived by Perrin (23). C is a parameter that describes how strongly the solvent and solute molecules are attractively coupled. In the stick boundary condition, C = 1; in the slip boundary condition values of C depend upon solute geometry. Knowing C is crucial to relating rol to 7. The rotational diffusion coefficient of a nonspherical solute is described by a second rank diagonal tensor whose elements are D,, D,, and D,, where x , y , and z refer to the Cartesian coordinates of the solute. The diffusion coefficient D , is the average of these three elements. In the slip boundary condition, the relations between the lengths of the solute axes and C have been derived by Hu and Zwanzig for spheroids (24) and Youngren and Acrivos for ellipsoids (25). Perrin’s relations for the stick boundary condition, which yield istick, and Youngren and Acrivos’ relations for the slip boundary condition, which yield C, provide reasonable predictions of D,, D,, and D, for real solutes. For systems described well by these relations, the rotational diffusion behavior is said to be hydrodynamic. Nonhydrodynamic behavior, where C is significantlysmaller than that predicted from slip hydrodynamics, has been observed for long-chain n-alkanes and n-alcohols (26-28). This “subslip” behavior is potentially a disastrous limitation to the study of chromatographic media because these media are often comprised of long alkyl chains. If subslip reorientation were to occur, the calculated value of the viscosity would be much lower than the actual value. A method of sensing subslip is needed to avoid this problem. Recently, our group studied the components of the rotational diffusion tensor of tetracene under subslip conditions (29). The results showed that subslip behavior is associated with values of D,lD, D y / D , and D J D that are different from the hydrodynamic predictions. This phenomenon was interpreted as being a consequence of solvent structuring, which is known to be significant for n-alkyl chains (30-35). The experimentally determined values of D J D , D,/D, and D,/D thus serve as a diagnostic for hydrodynamic behavior. To measure D spectroscopically, the correlation function of eq 2 has been solved for the diffusional boundary condition
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
148
after the equilibration period to confirm the integrity of tetracene as the fluorophor and the absence of any decomposition products.
1
I
X: 11.5
A
Y: 4 . 9
A z: 3 . 0 A
Figure 1 . Coordinate system of tetracene and hydrodynamic dimen-
sions.
by Chuang and Eisenthal (36). The resulting expression is a five-component exponential decay. For planar, symmetric solutes the expression simplifies to a double-exponential decay. For the case of tetracene the double-exponential decay is the following:
r ( t ) = 0.3(cos2 8 - y3 - (D,
+ D, cos2 8 + D,
sin2 B -
2 0 ) / A ) exp(-(6D - 2 A ) t ) + 0.3(cos2 0 - y3 + (D, + D, cos2 8 + D,, sin2 9 - 2D)/A) exp(-(6D 2A)t) (4)
+
Figure 1 shows the structure of tetracene, the designation of the molecular axes, and the estimated hydrodynamic dimensions, the last two of which were calculated by Huppert e t al. (37). The excitation transition moment coincides with the y axis, and the emission transition moment is a t an angle 0 with respect to the y axis. The parameter 1is a function of D,, D,, and D,. A = (Ox2+ DY2+ DZ2- D,D,
-
D,D, - DSD2)'I2(5)
Experimental measurement of the double-exponential decay parameters allows calculation of D by the sum of the inverse of the time constants. In addition to the time constants, quantitation of the preexponential factors allows calculation of the values of D,/D, D,,/D, and D,/D. Often, it is observed that one of the preexponential factors is approximately zero, preventing determination of D,/D, D,.lD, and D,/D. Tetracene is a useful probe solute because it exhibits doubleexponential behavior.
EXPERIMENTAL SECTION Sodium dodecyl sulfate (SDS) was purchased from Aldrich in 98% purity. In preliminary studies, no change in behavior was observed for SDS purified either by recrystallization or by a chromatographic procedure (38). Therefme, SDS was used without further purification. Tetracene was purchased from Aldrich, and n-propanol, from Fisher. Tetracene's purity was tested with high-performance liquid chromatography, which showed negligible impurities, and was then used without further purification. The solutions containing n-propanol were made by preparing a stock solution of tetracene in n-propanol and adding a constant volume of the stock solution and the appropriate additional volume of n-propanol to a 0.1 M SDS solution. The concentration of tetracene in all cases was 0.2 pM. For the solution containing no n-propanol, solid tetracene was dissolved into the micellar solution by stirring until the resulting fluorescence intensity was the same as for the solutions containing n-propanol; this required 10 h. On the basis of -60 SDS monomers/micelle and the known critical micelle concentrations as a function of propanol content (39),the number of tetracene molecules/micelle averaged less than 1.5 X thus avoiding the effects of multiple occupancy of a micelle by tetracene. The concentration of tetracene was sufficiently high that the blank contribution was negligible. All solutions were allowed to equilibrate at room temperature for l week because long equilibration times for micellar solutions have been reported ( 3 0 , 4 2 ) . Fluorescence emission spectra were obtained
The solutions were not deoxygenated. The temperature was controlled at 25 f 0.5 "C for the fluorescence depolarization measurements. The 476-nm line of an argon ion laser was used for exciting the samples, and the power was 10 mW. The fluorescence emission of tetracene was isolated at 512 nm by using a monochromator with a 2-nm bandwidth. A fixed Polaroid was placed at the entrance to the monochromator. The spectroscopic equipment used for this work is identical with that used in a previous experiment (29) where the lst, 2nd, 4th, 7th, and 9th harmonics of the 82-MHz mode beat frequency of a mode-locked argon ion laser were used as the sinusoidal modulations. For each frequency, two measurements are made: A@,,- AbL and al,/ai, which are the differential phase shifts and demodulation ratios that comprise r ( t ) (19, 20). The instrumental uncertainties in the phase and amplitude calibrations were combined with the noise in the experiment for the total experimental variance used in the regression of the data. To avoid false minima in the data analysis, a mainframe computer was used to calculate the phase shifts and demodulation ratios for a large set spanning all possible decay parameters, and the global minimum of x2 value was identified. This makes use of the fact that the theoretically possible answers are bounded, i.e., 0 5 r(0) 5 0 . 4 , 0 I T~~ 5 8 ns, where 8 ns is estimated to be the reorientation time of the micelle (19), and 0 5 F 5 1, where F is the fractional contribution of each exponential. After the global minimum was found, a finer increment was used, if necessary, to find the local minimum.
RESULTS AND DISCUSSION The raw frequency-domain data are shown graphically in Figure 2. The differential phase shift A4,, - AdL and the amplitude ratio a , , / aI both behave normally for each solution, showing their expected increases with frequency. Definite differences in the reorientation behaviors of the solutions are revealed as the n-propanol concentration is varied: the changes in the values of and a 8 , / a Lexceed the 95% confidence intervals from one solution to the next. In frequency-domain experiments, if other factors such as transition symmetry and solvation structure are constant, larger phase shifts and higher amplitude ratios correspond to longer reorientation times. The raw data therefore suggest that the addition of n-propanol causes faster reorientation of tetracene. It is important to ensure that the changes in the frequency-domain data upon addition of n-propanol are not due simply to an increased concentration of tetracene in the less viscous, aqueous interstitial solution. The composition of the interstitial solution was mimicked by preparing an SDS solution just below the critical micelle concentration. To represent the worst possible interference, a saturated solution of tetracene in this medium was prepared by equilibration with solid tetracene for 1 day, and 10% n-propanol was used to maximize solubility. The fluorescence emission spectrum of tetracene in this premicellar solution is shown in Figure 3, in comparison to the emission spectrum of tetracene in the micellar solution. Very little emission is detected from the premicellar solution, and the emission is broad and shifted to the red of the emission from the micellar solution. This premicellar solution, saturated in tetracene, contributes less than 2 % of the signal from the micellar solution a t 512 nm. In frequency-domain experiments, which are sensitive to the areas under the decay curves, this contribution would be less than 0.3%, based upon the viscosity of 1 CPfor water. In the actual micellar solution, the interstitial solution would not be saturated; it would compete with the micelles for the limited amount of tetracene, thus contributing even less to the signal. It is concluded that the fluorescence emission of tetracene in the micellar solutions has negligible contribution from tetracene in the interstitial solution. T o interpret the experimental results quantitatively, the data sets of Figure 2 are fit to the equation relating A 4 - A@J.
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
T"
20.0
Table I. Anisotropy Decay Parameters of Tetracene in SDS Micelles as a Function of n -Propanol Concentration'
- 1.0%
t
1.5%
70 PrOH
3.0%
0
:I
10
a
1.5
a
b b 3.0
r
0
10.0
0'8.00
do.
I
I
I
I
'
I
I I
I
I
700.
300. 500. I Frequency IMHzI
I
3"
I I
I
9do.
i
D , no PrOH
,i
1
"r 1.29
51.21
0
