J . Phys. Chem. 1989, 93, 7730-7732
7730
Electrk Birefringence at Small Analyzer Orlentation Angles Nancy Stellwagen Department of Biochemistry, University of Iowa, Iowa City, Iowa 52242 (Received: March 1, 1989; In Final Form: April 21, 1989)
The amplitude of the electric birefringence of a variety of solutions has been measured as a function of electric field strength and the orientation of the analyzer about the crossed position. The Kerr constants of water and propylene carbonate were calculated from the variation of the optical retardation, 6, as a function of analyzer orientation at constant electric field strength and from the variation of 6 as a function of electric field strength at constant analyzer orientation. Both methods give equivalent results. It is also shown that the minimum of the steady-state birefringence occurs not at the crossed position, but at an analyzer orientation corresponding t o -614. An explanation is offered for this effect.
Introduction Electric birefringence is one of the few techniques that is able to measure the orientation of macromolecules in solution in response to an applied electric field. Many different types of macromolecules, including proteins, DNA, polynucleotides, and synthetic polymers, have been studied by this technique.' Electric birefringence is increasingly being used to study orientation in unusual situations, such as the orientation of DNA in agarose and poly(acry1amide) Therefore, it is important to understand the technique thoroughly, in order to be able to interpret the data with confidence. In an electric birefringence apparatus using a quarter wave plate to enable linear, instead of quadratic, detection of the birefringence signa1,6+'two methods can be used to measure the Kerr constant of the sample under study. In the most widely used method, the amplitude of the birefringence is measured as a function of electric field strength',' and the Kerr constant calculated from the Kerr law:
An = n,, - n, = KnE2
(1)
Here, An is the birefringence, the difference in refractive index parallel and perpendicular to the direction of the applied electric field, n is the mean refractive index of the solution, E is the electric field strength, and K is a proportionality constant, known as the Kerr constant, which is characteristic of the material under study. A second method to obtain the Kerr constant is to measure the amplitude of the birefringence as a function of the rotation of the analyzer from the crossed position at constant electric field strength.8-'0 This method is less well-known but has been used as the basis of a signal-nulling technique for measuring birefringence." In this report the electric birefringence of water, propylene carbonate, and an aqueous solution of DNA has been measured as a function of electric field strength and analyzer orientation angle. Both methods give equivalent results. Contrary to previous ( I ) Reviews of the literature are given by: OKonski, C. T. Encycl. Polym. Sci. Technol. 1968, 9, 551-590. Stoylov, S. P. Adv. Colloid Interface Sci. 1971, 3, 45-1 10. OKonski, C. T., Ed. Molecular Optics; Marcel Dekker: New York, 1976, 1978; Parts 1 and 2. Charney, E. Q. Rev. Eiophys. 1988, 21, 1-60. Useful compilations are also given in the following volumes: Jennings, B. R., Ed. Electro-Optics and Dielectrics of Macromolecules and Colloids; Plenum Press: New York, 1979. Krause, S. Molecular ElectroOptics; Plenum Press: New York, 1981. (2) Stellwagen, N . C. J. Eiomol. Strucr. Dyn. 1985, 3, 299-314. (3) Wijmenga, S. S.; Maxwell, A. Biopolymers 1986, 25, 2173-2186. (4) Stellwagen, N. C. Biochemistry 1988, 27, 6417-6424. (5) Chu, B.; Xu, R.; Wang, 2.Biopolymers 1988, 27, 2005-2009. (6) OKonski, C. T.; Zimm, B. H. Science 1950, 111, 113-116. (7) Frcdericq, E.; Houssier, C. Electric Dichroism and Electric Birefringence; Clarendon Press: Oxford, 1973. (8) Kikuchi, K.; Yoshioka, K. J . Phys. Chem. 1973, 77, 2101-2107. (9) Ravey, J. C.; Houssier, C. In Elecrro-Optics and Dielectrics of Macromolecules and Colloids; Plenum Press: New York, 1979; pp 67-76. (10) Houssier, C.; OKonski, C. T. In Molecular Electro-Optics; Krause, S.,Ed.; Plenum Press: New York, 1981; pp 309-339. ( 1 I ) Beevers, M. S.; Khanarian, G . Aust. J . Chem. 1979, 32, 263-269.
0022-3654/89/2093-7730$01.50/0
treatments?*l0the minimum absolute value of the birefringence signal is found to occur not a t the crossed position, but at an analyzer orientation of -6/4, where b is the phase retardation of the sample under the given experimental conditions. An explanation is offered for this effect. The Kerr constants calculated for propylene carbonate and water are also compared with other values in the literature. Materials and Methods Materials. Propylene carbonate was obtained from Eastman Kodak Chemicals and used without further purification. A 1426 base pair (bp) DNA restriction fragment was isolated from plasmid pBR322 as described previously.I2 Stock solutions of the fragment were stored in a freezer in TO.1E buffer (10 mM Tris-HC1, pH 7.8, plus 0.1 mM EDTA) until needed; the solutions were stable for months. Immediately before each experiment, an aliquot of the DNA stock solution was diluted to a concentration of 10-20 pg/mL in the desired buffer. The water used to prepare all solutions was 18 Megaohm distilled deionized water, obtained from a Barnstad NANOpure deionizer just prior to use. All chemicals were reagent grade. Apparatus and Methods. The electric birefringence apparatus used in the present measurements has been described brieflya2 The light source was a Spectra Physics Model 145P 1.5-mW He-Ne laser (A = 633 nm). The polarizer and analyzer were GlanThompson prisms (Karl Lambrecht Corp.), mounted in drum-type rotators with vernier scales which could be read to 0.1O. The optical system included a quarter wave plate in front of the analyzer to permit linear detection of the birefringence signals. Light intensities were converted into voltage by a photodiode, amplified by optical amplifiers, digitized by a Nicolet Model 2090 oscilloscope equipped with Model 140A dual-channel plug-in unit, and stored with or without signal averaging by an Altos Model 8000-2 computer. Graphic displays of the oscilloscope traces were recorded on an Anadex Model DP9501 printer. Pulses were generated by a Cober Electronics (Stamford, CN) Model 605P high-power pulser. The polarity of the pulse was reversed between pulses by a Kilovac (Pasadena, CA) high-voltage switch, in order to prevent electrophoresis and/or electrode polarization. The Kerr cell was a shortened 1.00-cm quartz spectrophotometer cell, chosen for its negligible strain birefringence. The same cell was used for all measurements reported here. A 400-ohm low-impedance resistor was connected in parallel with the cell to keep the current across the cell constant. Using a 200-ohm resistor had no effect on the results. The electrodes were parallel platinum plates with 2.1- or 1.5-mm electrode separations, mounted on lexan supports of standard design.13 The cell was placed in a brass cell mount thermostated at 20.0 "C. The temperature in the cell was measured with an Omega digital thermometer, Model 450 AKT. The stray light constant7J4 of the apparatus, with cell and elec(12) Stellwagen, N. C. Biochemistry 1984, 23, 6311-6319. (1 3) Pytkowicz, R. M.; O'Konski, C. T. Biochim. Eiophys. Acta 1959,36, 466-470.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 22, 1989 7731
Electric Birefringence of Solutions IAVI. mV
Propylene Carbonate
Figure 1. A plot of the absolute value of AV vs a for the 1426 bp DNA restriction fragment. DNA concentration = 14.3 pg/mL in 0.2 mM Tris buffer. E = 3.51 kV/cm, 6 = -1.32"; minimum occurs at a = +0.31". AVcan be used as the ordinate instead of the left side of eq 4 because the stray light constant is small. If the actual values of AVwere plotted instead of I A Q the right arm of the curve would be shifted into the lower right quadrant, since a decrease in light density is observed at positive
analyzer orientations for samples that are negatively birefringent.
The time constant of the
trodes in place, was (0.5-1.0) X detecting circuit was -0.2 ps. Theory of the Measurements. The electric birefringence was measured in terms of the phase retardation, 6, induced in each solution, according to 6=- 2rAnl
x
where X is the wavelength of the incident light, An is the birefringence, and I is the path length of the cell. The exact equation of an optical system employing a quarter wave plate with slow axis oriented at 1 3 5 O with respect to the direction of the applied electric field is7
where AI is the change in light intensity caused by application of the electric field, I, is the light intensity observed with the analyzer rotated a degrees from the position crossed to the polarizer, AV and Vu are the corresponding output voltages from the detector, 6 is the phase retardation of the sample, and KSL is the stray light c o n ~ t a n t . ~ .In ' ~ eq 3, the strain birefringence of the cell is assumed to be negligible, as true for the cell used in the present studies. The Kerr constant of the sample is calculated from the measured values of AV and Vu and eq 1-3. Rearranging eq 3 for small values of a and 6 (AV/V,)(sinz a
-
+ KsL)
-
a6
+ b2/4
-
ab
(4)
since sin x x for small x and the term in 62/4 is usually much smaller than ab. Hence, 6 can be calculated from the slope of the straight line obtained when the left side of eq 4 is plotted as a function of a. Plots of eq 4 have also been used to determine the strain birefringence of the birefringence cell,9 assuming that the minimum in the birefringence signal occurs at the crossed position where a = 0.lo However, at very small values of a,the term in 62/4 becomes comparable to a6,and the minimum in the birefringence signal will occur not at a = 0 but at a = -614. This result is demonstrated below.
Results and Discussion Typical electric birefringence data for a 1426 bp DNA restriction fragment plotted in the manner suggested by eq 4 are shown in Figure 1 . Since the stray light constant was small and the response of the optical system was linear, the left side of eq (14) Stellwagen, N. C. Ph.D. Thesis, University of 1967.
California, Berkeley,
+
25
Figure 2. A plot of the absolute value of (AV/V,)(sin2 a + KsL) vs a for propylene carbonate. E = 5.57 kV/cm, 6 = +1.22'; minimum occurs at a = -0.33". Here the absolute value of the left side of eq 4 is plotted as the ordinate. If the actual values of (AV/V,)(sin* a + KsL)were
plotted, the left arm of the curve would be shifted into the lower left quadrant, as in Figure 3.
4 can be simplified to AV. Following the traditional manner of presentation?*10the absolute value of AV is plotted as a function of a in Figure 1. It can easily be seen that the minimum in the steady-state birefringence occurred not at the crossed position, but at a = +0.31'. The retardation of this solution was calculated to be -1.32O from the slope of the line in Figure 1, and also from eq 1 and the experimentally determined values of 6 and E . Therefore, within experimental error, the analyzer orientation corresponding to zero birefringence was equal to a = -614. By use of the same cell, typical electric birefringence data obtained for propylene carbonate are shown in Figure 2. Propylene carbonate is a useful liquid to study because it has a very large Kerr ~ o n s t a n t ' and ~ . ~ is~ stable and relatively nontoxic. In this figure the absolute value of the left side of eq 4 was plotted as a function of analyzer orientation. The minimum in the steady-state birefringence signal was found to occur at 6 = -0.33'. The retardation of propylene carbonate at this electric field strength, calculated from the slope of the line in Figure 2, was 1.22'. Again, within experimental error, zero birefringence occurred at an analyzer orientation of a = -614. Since the same cell was used for the measurements reported in Figures 1 and 2, the position of the minimum in the steady-state birefringence is unrelated to the strain birefringence of the cell; instead, the minimum occurs at an analyzer orientation of a = -614. This relationship was predicted from the discussion of eq 4 above. Since 6 increases with increasing electric field strength, as shown by eq 1, the analyzer orientation corresponding to zero birefringence should also increase with increasing field strength. This dependence is illustrated in Figure 3. Here the actual value of AV is plotted as a function of a. The analyzer orientation corresponding to the minimum birefringence increased from a = -0.15' to a = -0.52' as E increased from 3.61 to 7.79 kV/cm. The slopes of the lines also increased with increasing electric field strength, as predicted by eq 4. The slopes of the lines in Figures 1-3 can be used to calculate the Kerr constants of the various samples, following the treatments of Kikuchi and Yoshioka: Ravey and Houssier? and Beeves and Khanarian." For propylene carbonate, the slopes of the lines in Figures 2 and 3 correspond to a Kerr constant, K of (4.3 f 0.1) X cgs esu (4.8 X mks). A log-log plot of the birefringence of propylene carbonate as a function of electric field strength is shown in Figure 4. The slope of the line is 2.01, indicating that the Kerr law, eq 1, is obeyed. The value of the Kerr constant, K , calculated from this data is (4.3 f 0.3) X lo-'' cgs esu (4.8 X mks), in excellent agreement with the value calculated from the slopes of the lines in Figures 2 and 3. These values are compared with literature values in Table I. The literature values were recalculated from the reported values of
+
(15) Krause,
S.Ph.D. Thesis, University of California, Berkeley, 1955.
7732 The Journal of Physical Chemistry, Vol. 93, No. 22, 1989
10 mM N r
"-
1500
I I
*"""
B
Stellwagen 10L
b = 2 12'
; 1; -
4
b
1.o
deg.
0.10
Propylene Carbonate
L
I
IIIIII
1
/
A
I
t
Slope = 2 01
L
I
I
, 1 1 1 1 1
10
Figure 4. log-log plot of the retardation of propylene carbonate as a function of electric field strength. The slope of the drawn line is 2.01, indicating that the Kerr law is obeyed. -1500
TABLE II: Kerr Constant of Water
Figure 3. AVvs a for a solution of 10 m M NaI in propylene carbonate measured at three different values of the applied electric field strength: (a) E = 3.61 kV/cm, 6 = +0.47'; (b) E = 5.57 kV/cm, 6 = +1.16'; (c) E = 7.79kVJcm, 6 = +2.12'. For these three cases, zero birefringence is observed at (a) a = -0.15', (b) a = -0.33", and (c) a = -0.52', respectively. The values predicted from a = -6/4 are -0.12', -0.29', and -0.53' for (a), (b), and (c), respectively. The data are plotted with A V as the ordinate to illustrate the equivalency of the results with Figure 2. To simplify the figure, the data are plotted as the actual change in voltage at each analyzer orientation.
TABLE I: Kerr Constant of Propylwe Carbonate A, B x 106, K x 1010, nm cgs esu cgs esu 13.9 5.2 KrauseI5 540 Stellwagen14 -510 12.6 4.5 this work (6 vs E) 633 9.86 4.3 4.3 633 9.g5 this work (6 vs a)
K x 1019, mks 5.8 5.0 4.8 4.8
B, another Kerr constant sometimes used for pure liquids, by using the relation'
K = BX/n
(5)
The refractive index of propylene carbonate was taken to be 1 .439.16 A sensitive test of the accuracy of a birefringence instrument is to measure the Ken constant of water, which is 50 times smaller than that of propylene carbonate. Therefore, the Kerr constant of water was measured by the two methods described here, varying the electric field strength at constant analyzer orientation and varying the analyzer orientation at constant electric field strength. More than 20 separate determinations of the Kerr constant of water were made, using different cells and different electrode assemblies. The average value of the Kerr constant, K, determined cgs esu (1.40 X from plots of 6 vs E2 was (12.6 f 0.7) X mks); the average value of K determined from plots of 6 vs LY at constant E was (12.2 f 0.3) X lo-'* cgs esu (1.36 X mks). These two values agree within experimental error. These values of the Kerr constant of water are compared with other values in the literature in Table 11. Where necessary, Kerr (16)Horne, R.A., Ed. Water and Aqueous Solutions; Structure, Thermodynamics and Transport Processes; Wiley-Interscience: New York, 1972; p 112.
' "'''$j '
'
/d
;j"
:
, ,,,,,(
B x 107, K X 1Ol2, K X lozo, cgs esu cgs esu mks 540 2.43 9.8 1.09 436 3.12 12.2 1.35 2.89 11.9 1.32 546 578 2.72 11.8 1.31 2.83 10.8 -510 1.20 16.2 1.8 550 12.8 2.79 1.42 633 514.5 3.10 13.3 1.48 633 1.80 16.2 633 2.64 1.40 12.6 633 2.56 1.36 12.2 A,
nm
Krause" Orttung et a1.I8 Orttung et a l l 8 Orttung et a1.18 Stel1wagenl4 Tricot et aI.I9 Khanarian and Kentz0 Elias and Edenz1 Wijmenga and Maxwell' this work (6 vs E) this work (6 vs a)
constants were calculated from the reported values of B via eq 5. For pure liquids, which obey the Kerr law over the range of electric field strengths commonly used in electric birefringence experiments, it is equally valid to determine the Kerr constant from the variation of the analyzer orientation at constant electric field strength or from the variation of electric field strength at constant analyzer orientation. Of course, determining the Kerr constant by using both methods would minimize the chance that a small error in the measurement of either electric field strength or analyzer orientation would unduly influence the results. The Kerr constants of macromolecules are better obtained from the variation of the retardation with electric field strength, since saturation of the birefringence can occur in high electric fields and the rate of approach of the birefringenw to its saturation value can be interpreted in terms of the mechanism of orientation of the macromolecule in the electric field." Acknowledgment. The assistance of Mr. Craig Fastenow (University of Iowa, Department of Biomedical Engineering) in designing the optical amplifier is gratefully acknowledged. Financial support from Grant GM-29690 from the National Institutes of General Medical Sciences is also acknowledged. Registry No. H20, 7732-18-5;propylene carbonate, 108-32-7. (17)OKonski, C.T.;Yoshioka, K.; Orttung, W. H. J. Phys. Chem. 1959, 63, 1558-1565. (18)Orttung, W.H.; Meyers, J. A. J . Phys. Chem. 1963,67, 1905-1910. (19)Tricot, M.; Houssier, C.; Desreux, V . Eur. Polym. J . 1978, 14, 307-315. (20) Khanarian, G.;Kent, L. J . Chem. Soc., Faraday Trans. 2 1981,77, 495-501. (21)Elias, J. G.;Eden, D. Macromolecules 1981, f4, 410-419.
