reference, the following would be appropriate for a set of peaks that do not differ too much in kinetic energy (and thus also attenuation length): 1) A larger enhancement ratio indicates that a given species is distributed nearer the surface. 2) A ratio > unity indicates some degree of concentration near the surface. 3) A ratio very near unity suggests a species homogeneously distributed in the substrate. 4) A ratio < unity suggests a species whose maximum concentration may be somewhat below the surface of the substrate. For a group of peaks with a lar,ge kinetic energy range, the interpretation of subtle changes in enhancement ratio may be difficult, however. For example, the 0 Is ratio is consistently slightly less than unity in the present data as well as in that of other similar studies of A1203 powder (121, and the normalized 0 ls/Al 2p(oxide) ratio for the aluminum specimen of Figure 3 is also somewhat less than unity a t 10'. These effects could be due to the lower attenuation length for 0 Is photoelectrons (kinetic energy = 955 eV) as compared to the reference A1 2s photoelectrons (kinetic energy = 1365 eV), which would cause more attenuation of 0 Is in any overlayers present on the oxide; such differences in attenuation would be amplified a t lower angles. Alternatively, however, these enhancement ratios slightly less than unity could be due to a slight reduction in the O/Al concentration ratios near the surface. One test for the energy-dependent attenuation length effect might consist of enhancement ratio measurements for the low-lying band of 0 2s states observed in the valence region of A1203, which possesses a kinetic energy greater than that of A1 2s. This band, however, is much lower in intensity than 0 Is and thus difficult to measure with sufficient precision, and
also is subject to certain ambiguities of interpretation because of the possibility of 0 2s-derived levels for other near-surface species which could occur a t somewhat different energies.
LITERATURE CITED
4.
(1) C. S.Fadley and S. L. Bergstrom, Phys. Lett. A, 35, 375 (1971). (2) C. S.Fadley and S.A. L. Bergstrom, in "Electron Spectroscopy", D. A. Shirley, Ed., North Holland Publishing Co., 1972, p 233. (3) W. A. Fraser, J. V. Florio, W. N. Delgass, and W. D. Robertson, Surf. Sci., 36,661 (1973). (4) C. S. Fadley, R. J. Baird, W. Siekhaus, T. Novakov, and S. L. Bergstrom, J. Electron Spectrosc., 4, 93 (1974). (5) C. S. Fadley, J. Electron Spectrosc., 5, 725 (1974). (6) J. Brunner and H. Zogg. J. Electron Spectrosc., 5, 81 1 (1974). (7) R. J. Baird, C. S.Fadley, S.K. Kawamoto, and M. Mehta, Chem. Phys. Lett., 34, 49 (1975). ( 8 ) C. S.Fadley, to appear in Faraday Society Discussion, NO. 60. (9) R . J. Baird and C. S. Fadley. abstract for the 30th Northwest Regional Meeting, ACS, Honolulu, Hawaii, June 12-13, 1975, and unpublished results. (10) The AB Metal Digest, Vol. 11, No. 213 (1973), Buehler Ltd., Evanston, Ill. (11) J. E. Boggioand R. C. Plumb, J. Chem. Phys., 44, 1081 (1966). (12) R. Alvarez, Ph.D. Thesis, Department of Agronomy and Soil Science, University of Hawaii, 1975, and R . Aivarez, C. S.Fadley, J. A. Silva, and G. Uehara, unpublished results. (13) B. L. Henke and M. Tester in "Advances in X-ray Analysis". Vol. 18, Plenum Press, New York, 1975.
A.
RECEIVEDfor review September 3,1975. Accepted January 12, 1976. This work was presented in part a t the Faraday Discussion on Electron Spectroscopy of Solids and Surfaces, Vancouver, Canada, July 15-18, 1975. Those portions of the study performed in the Chemistry Department, University of Hawaii, were also supported by the National Science Foundation (Grant GP 38640X) and the University of Hawaii Research Council.
Digital, Photon-Counting Fluorescence Polarometer I?.J. Kelly and W. B. Dandliker" Department o f Biochemistry, Scripps Clinic and Research Foundation, La Jolla, Calif. 92037
Donald E. Wllliamson Cordis Corporation, Miami, Fla. 33 127
The design and construction features of a digital, photoncounting instrument for the measurement of fluorescence polarization (fluorescence poiarometer) are described. The instrument also performs the functions of a highly sensitive, stable fluorometer. The data from photon counts is processed in a digital computer and is displayed digitally. The design features of the instrument make It possible to achieve very high stability and sensltlvlty, while maintaining low levels of illumination of the sample. The standard error of estimate in making polarization measurements is fO.OO1 polarization unit. As a fluorometer, the response is linear over a wide dynamic range and useful measurements M fluorescein. can be made down to about 5 X
The dependence of the polarization of fluorescence upon rotational relaxation times provides information on the size, shape, and segmental motion of macromolecules in SO846
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
lution. In addition, fluorescence polarization methods lend themselves to the measurement of the kinetic and equilibrium parameters of reactions over extremely wide ranges of concentration when a fluorescent or fluorescence-labeled molecule is involved ( I ). Also, polarization measurements can be readily adapted to give sensitive methods of assay for many substances present only a t a few ng/ml(2). Instrumentation is a crucial factor in making fluorescence polarization measurements of sufficient precision to realize the full potential of the method in the kinetic, equilibrium, and assay areas. A number of fluorescence polarometers (instruments for measuring fluorescence polarization) with a variety of electronic and optical arrangements have been described in the past decade. In our early instruments ( 3 ) , the fluorescence intensity was compared to that of a reference beam by means of either an Aryton shunt and galvanometer or of a synchronous chopper and tuned amplifier. Monnerie and NBel ( 4 ) employed a galvanometer readout and servoed the photomultiplier with a signal from
a reference beam. Deranleau ( 5 ) employed an ingenious optical means of obtaining a reference signal and processed the signals from two photomultipliers by analog computer to record polarization or intensity as functions of wavelength or time. Aurich and Lippert (6) designed an instrument to measure fluorescence polarization spectra with special emphasis on the accuracy of the absolute values of the polarization. Weber and Bablouzian (7) utilized separate photomultipliers for the two polarized components and recorded the ratio as a function of wavelength. A very versatile instrument utilizing a beam splitter consisting of three quartz plates suitably placed and oriented was described by Witholt and Brand (8). McKay’s (9) fluorescence emission and polarization spectrometer employed a motor-driven polarizer for obtaining polarization measurements. Rosen (10) utilized two photomultipliers feeding into a solid state analog divider. Claesson and Odani ( 1 1 ) described an apparatus to measure polarization of a fluid under shear. Lavorel et al. (12) had an excellent arrangement with a rotating polarizer in the incident beam. Spencer e t al. ( 1 3 ) utilized analog detection but digital readout. Curry et al. ( 1 4 ) applied photon counting and pulse height discrimination in constructing a filter fluorometer. The recording polarization spectrometer of Wampler and De Sa ( 1 5 ) employed a piezoelectric birefringence modulator and an analyzing polarizer to generate a time-dependent intensity which is a function of the polarization. In the past, it has been rarely possible to compare the sensitivity and performance of fluorescence polarometers, partly because there has been no widely accepted way of defining sensitivity. A definition of sensitivity must take into account how the polarization, p , is measured and computed. T o facilitate the discussion assume that the incident beam, usually polarized in the z direction, is propagated along the 3: axis and the fluorescent light is observed along the y axis. The polarized components (or their intensities) in the fluorescent light can then be denoted by V (electric vector in the z direction) and by H (electric vector in the x direction. In many of the instruments discussed above, it appears that p was obtained directly from the V and H signals arising from the sample itself. This procedure is a satisfactory approximation a t high levels of fluorescence; but, a t low levels, the correction for the signals from the blank (no fluorescent component added) becomes very important and must be allowed for (Dandliker e t al.) (16).We defined the polarization in terms of an “excess fluorescence” indicated by A which is the difference between intensities for sample and blank. The polarization was computed from the excess fluorescence by the relationship A V - AH = A V + AH
(1)
Viewed in the most simple terms, the ultimate useful sensitivity attainable with an instrument depends upon the precision with which the signal arising from the blank itself can be measured. If the solutions and instrumentation were perfectly stable, then the ultimate limit on sensitivity would be imposed by the statistical nature of photoelectron emission. However, this factor can be minimized to any desired degree by lengthening the counting interval (See Appendix). Hence, from a practical viewpoint, the actual limits on sensitivity are determined by the reproducibility of the reading for the blank. Obviously, the situation is optimized by making the ratio of the reading for some arbitrarily chosen fluorescent solution to that of the blank, a maximum. From these considerations, a definition of sensitivity useful for comparing the performance of different instruments in different laboratories can be expressed in terms of the concentration of fluorescein (or other fluorescent mate-
rial) to which the uncertainty (e.g., standard error) in the reading of the blank is equivalent. While the exact form for expressing this uncertainty is arbitrary, twice the standard deviation in determining the intensity of the blank seems to be a reasonable choice. Thus, if the blank itself gives an intensity ( V H )reading (fluorescein equivalent) equivaM fluolent to the excess fluorescence observed for rescein and the standard deviation of the determinations for the blank is equivalent to M fluorescein, then the sensitivity of the instrument would be 2 X M fluorescein.
+
INSTRUMENT DESIGN Basic Design Goals. The instrument design described in this paper evolved from a need for high sensitivity, precision, and linearity, together with a minimal illumination of the sample and ease of operation a t one selected wavelength a t a time. T o attain these goals, the following design features were incorporated: 1) A detector utilizing only one photomultiplier tube. 2) Use of the detecting photomultiplier only as a photon counter with pulse height discrimination. 3) Digital processing and readout of the photon count. 4) A continuously rotating Polaroid analyzer making the measurements of p less susceptible to time-dependent variations in the signal, detector sensitivity, and excitation intensity. 5 ) A light source consisting of a tungsten lamp, the output of which is adjustable to f l %over a 500-fold range, and stabilized by feedback control. 6) Isolation of the excitation wavelengths and rejection of these wavelengths by a series of interference and absorption filters. 7 ) Thermostating of the cell holder (10.2 O C ) by means of a thermistor-controlled Peltier cooler. Optical Design. The arrangement of the optical components is shown in Figure 1. A tungsten lamp was chosen as the light source since the high intensities achievable with arc sources are somewhat undesirable from the viewpoint of local heating and possible photochemical changes but, more importantly, tungsten sources are not subject to erratic movement and stability of output is readily achieved. The lamp and blower are housed separately to avoid overheating interference filters Fa and F3. Interference filters used in tandem with each other or with nonfluorescent glass absorbing filters can achieve very high rejection of the incident wavelength, which is necessary to attain low blanks. In the incident beam, for example, an interference filter transmitting 62% a t 489 nm, transmits about 0.01% a t 520 nm (ratio = 6200). If two such filters are used in series, the ratio is 3.8 X lo7. The same considerations apply to the fluorescent beam. Optimal filter combinations are best chosen empirically by comparing the instrument response to a solution of the fluorescent dye to that of a suspension of Ludox (colloidal Silica from E. I. du Pont de Nemours, Wilmington, Del.) or other material which scatters but shows negligible fluorescence. The ratio of fluorescent readinghcattering reading should be maximized, while retaining sufficient signal for easy detection. T h e plate, P (Figure 11, is a 2-in. x %in. square of Pyrex glass (Corning Glassworks, Corning, N.Y., Glass No. 7740) which transmits down to about 300 nm. Low fluorescence quartz would be somewhat preferable. A simple vacuum phototube was chosen to monitor the incident intensity because of its high stability and insensitivity to supply voltage. Thermostating of the cell by thermoelectric cooling is quite precise and avoids the practical difficulties associated with circulating liquids. Control of the temperature is accomplished by feedback from a thermistor in the cell holder, C. Details of similar arrangements are well-known, and have not been included in the description below. Lenses used in the optical system are glass, but should be examined and chosen ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
847
for low fluorescence. The cuvette should be of low-fluorescence quartz, and may be either mirrored or painted black on two of the outside surfaces. While the mirrored cells provide a factor of 3 advantage in intensity, the blackened cells are far more reproducible and stable, both over time and from cell to cell. Polarizing filters Fq and F7 (Figure 1) are from Polaroid “-22 film (Polaroid Corp., Cambridge, Mass.). For measurements below about 400 nm, other polarizers should be used (e.g., Polacoat Corp., Cincinnati, Ohio). Action of the rotating polarizer and the LED, LST assembly is discussed in the Appendix. Electronic Design. The components of the detector module are shown in Figure 2. The photomultiplier feeds pulses originating from single photons into the SSR 1120 amplifier/discriminator. Only those pulses exceeding a predetermined amplitude are passed by the discriminator. In this way, “dark current” pulses are largely eliminated since they are usually of smaller amplitude than those originating from photons. In addition, the photomultiplier is also fitted with a magnetic adapter which reduces the effective photocathode diameter from 1 cm to 2 mm, with a corresponding reduction in dark current. However, this reduction makes optical alignment more difficult and also decreases the amount of light collected. Use of a smaller photomultiplier possibly could achieve the same benefits as our arrangement. Use of the photomultiplier only as a photon counter also makes the resulting detection less sensitive to time-dependent changes in photomultiplier gain or in power supply voltage. The latter is normally set at 1150 V and is supplied by a model 412B high voltage power supply (John Fluke Mfg. Co., Inc., Seattle, Wash.). The detector module also interfaces the chopper signal from the rotating analyzer to the SSR 1110 by means of the circuit shown in Figure 3. The chopper signal directs the pulses from the SSR 1120 into one of the two high-speed accumulating registers of the SSR 1110, as described in the Appendix. At the end of each measurement, the SSR 1110 computes and displays the sum and difference of these registers, to three significant digits. The excitor module (Figure 4) accepts photocurrent from the excitation monitor (929 phototube of Figure 1) and compares it with an operator-selected reference voltage. The resulting control signal is used to drive the excitation lamp to deliver that intensity at which the excitation monitor signal equals the selected reference value. In addition, the module limits the maximum lamp current to 8 A. Three light-emitting diodes provide visible indication in the event that the excitation energy is less than or greater than the reference, and in the event the lamp current limit is activated. Schematics for the several parts of this module are given in Figures 5-8. The excitor module is powered by a regulated rtl5 V supply based on the Motorola MC1539 and MC1561 integrated regulators (Motorola Semiconductor Products, Inc., Phoenix, Ariz.).
EXPERIMENTAL The principal goals of the experiments reported here were to characterize instrument performance and utility and to verify the theory, presented in the Appendix, for the interpretation of measurements made with the rotating analyzer. Materials. The reagents used were saline-azide-phosphate (SAP) buffer (0.15 M NaC1, 0.01 M KzHP04, 0.005 M K H ~ P 0 4 , 0 . 4 g NaN3/1.), M fluorescein [33.2 mg mol) fluorescein (free acid form prepared according to Dandliker and Alonso, 1967) was dissolved in SAP and made up to 100 ml], and 55.5 wt. Oh sucrose in SAP. A fluorescent glass standard was made from a uranium glass standard, Type F53 (Carl Zeiss, Burbank, Calif.), intended for use with the Zeiss ZFM4 fluorometer. A dark filter consisting of a piece of Oxweld filter glass, shade AA-10, (Linde Division, Union Carbide, Long Beach, Calif.) was cemented onto the standard and attenuated the incident beam, thus bringing the observed fluorescence down to levels usable in our instrument. With 848
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
13
LEO F7
LST
I
Figure 1. Optical
10 cm
1
component layout (plan)
Light from the projection lamp (Sylvania DEF rated at 150 W, 21.5 V) passes through opening, S, (IO mm in diameter), heat filter F1 (Corning 1-69) and into lens L,, which focuses an image of the source into the fluorescence cuvette held in a copper block, C, which is thermostated by a Peltier cooler. The entire cuvette compartment is fitted with a light-tight lid. To protect the photomultiplier from ambient light, a shutter drops into the closed position when the lid is lifted and opens when the lid is replaced. Interference filters Fp and F3 isolate the desired wavelength (490 nm if fluorescein is to be used) and polarizer Fd (Polaroid film ”-22 or ”-38) transmits light with the electric vector perpendicular to the plane of the drawing. Quartz or glass plate, P, reflects a few percent of the incident beam onto the photocathode of a vacuum phototube (RCA type 929). which furnishes the reference signal for the lamp control circuit. Fluorescent light from the solution in the fiuorescence cuvette is focused by lens, L2, onto the rotating analyzer, F,. The fluorescence cuvette is of standard size (1-cm2 X 5 cm high) made of low fluorescence quartz and painted black on two of the outside surfaces. The incident wavelength is removed by interference filter F5 and colored glass F6 (chosen for very low fluorescence). if fluorescein is to be used, F5 is peaked at 520 nm and F6 is a selected piece of Corning glass 3-69. Rotating polarizer, F,. is made from a Polaroid sheet and is rotated at 90 rpm by a synchronous motor, M. The light-emitting diode, light-sensitive transistor (LED, LST) assembly, in conjunction with the light-chopping mask attached to the rotating polarizer (see Figure 9) furnishes a signal for phase discrimination in the phoion counting circuit. Light from the fluorescent solution passing through the rotating polarizer central to the mask mentioned above is focused by lens, Ls, near the plane of the photocathode of the photomultiplier tube (EM1 9635QB), which is housed in the photomultiplier housing PM (SSR Model 1151 from SSR Instruments, Santa Monica, Calif.)
the filter in place, the fluorescence of the glass standard is equivaM fluorescein in SAP. lent to approximately To obtain light showing zero polarization, needed to test instrument response, a solution of fluorescein was viewed by the photomultiplier through an opal glass filter acting as a depolarizer. If the insertion of a second opal glass does not further lower p , then it may be assumed that a single opal glass is sufficient to depolarize fully. Methods. 1) Measure fluorescence polarization and intensity. The instrument is warmed up for about an hour. During this time, the power supply for the lamp and power to the computer itself need not be on. The fluorescence cuvette and solution or the glass standard is then put into place and allowed to come to constant temperature. A light source intensity is selected to give perhaps u p to lo5 or lo6 countdsec and measurements are begun. The quantities read off the computer are SUM and DIFF, which are proportional respectively to V H,and to v(V - H ) , and N , the number of chopper cycles. The most simplified procedure for handling these data is as follows. Obtain, from the computer registers, SUM and DIFF for both sample and blank, keeping the value of SUM below l o 5 counts/sec so that dead time corrections can be neglected. Take the differences, solution minus blank, to obtain
+
1
_____
EMI 9 6 3 5 0 8 PHOTOMULTIPLIER
AMPLIFIER AND DISCRIMINATOR
",,"ciri
SY"%%OUS COMPUTER
L
I
~i F 1 rL-----J
EXCITOR LAMP CONTROL AMPLIFIER
i
POWER SUPPLY
Lr--.; j
1
REFERENCE VOLTAGE
I
i
1
Figure 2. Block diagram of the detector module The SSR 1110 computer, the SSR 1120 amplifierldiscriminator. as well as the photomultiplier housing (SSR 1151) were units purchased from SSR Instruments Co., Santa Monica, Calif. The chopper signal is furnished by a light-sensitive transistor viewing a light-emitting diode through the rotating polarizer and its associated mask (see Figures 1 and 9). The SSR 1110 displays, inter alia, two quantities, DlFF and SUM, which are the difference and sum, respectively, of photon counts over a selectable number of chopper cycles from the vertically-polarized ( V, and horizontally-polarized (Hj components in the fluorescent light. The polarization, p , is calculated from DlFF and SUM for the blank and sample according to Equation 2
+15V
INDICATOR ASSEMBLY
Figure 4. Block diagram of the excitor module A signal from the excitation monitor (the 929 phototube of Figure 1) is continuously compared to a standard reference voltage. The difference signal automatically maintains the lamp output constant at any one of 12 levels selectable over a 500-fold range of intensity
ICI
L
r\
x4
1K
2
-1
MLED60
5
+A A
EXCIToR LAMP
1
i l
Lamp
'lN914'
Figure 3. Schematic of the interface between the chopper (LED,
LST of Figure 1 ) and the digital synchronous computer (See Figure 2)
-Prt. Ext Mon
4
1
-k
rh Figure 5. Lamp control circuit (Module C)
U S U M ) and A(D1FF). The excess fluorescent intensity, A ( V is proportional to A(SUM), and the polarization is
+ H)
Values of 7 are given in Table IV, or can be calculated from Equation ix in the Appendix. Measurements made a t different numbers (N) of chopper cycles and different gate times (t,) may be normalized to counts/sec by dividing the observed number of counts for the interval taken by 2Nt,. In addition, measurements may be normalized to a unit value of LIF (lamp intensity factor) to facilitate comparison of data obtained a t different incident intensities.
RESULTS The performance of the instrument is demonstrated by the data presented in Tables I through V in which fluorescein solutions or a fluorescent glass standard were measured under a variety of instrumental conditions. Also, the same instrument has been used in connection with some of our previously-published work (Dandliker et al.) (2). The data of Tables I and I1 show that the theoretical model developed in the first section of the Appendix is highly consistent with experimental data. Thus, data on a particular solution obtained a t different incident intensities, number of chopper cycles, N , or at different gate times, t,, when normalized for these factors, all give the same final corrected value. Furthermore, data obtained as much as a month
Control is accomplished by the "current limit" circuit (Figure 6) connecting to points 0 and P and by the control signal feeding through R27 which controls the conductance of X16. This signal originates from the circuit shown in Figure 8 and enters at points D and E of Figure 5. A signal feeds out through point K to the circuit of Figure 7 which supplies the "over"/"under" indication. The power supply (Model CA-150, Illumination Industries Inc.. Sunny. vale, Calif.) for the lamp feeds in at V, and is capable of delivering up to about 10 A at 30 V dc. A variable transformer on the input of this supply is used to adjust V, to the correct range where control will ensue as indicated by both the "over" and "under" indicators (Figure 7)being off. The parts list includes A3 = 741 op amp Fairchild U6A7741; D3 = Silicon diode, 1N914; M = 1-mA panel meter, edge reading; R27, R31 = 1-KR. l o % , Ohmite RC20; R28-R30 = 49.9-KR, 1 % , Corning RN6OC; R32 = lO-KR, 5 0 % , Ohmite; R40 = 0.54,200-W (two 1 4 , 100-W. w.w, in parallel), Ohmite; R41 = 2.0-R, 100-W, w.w, Ohmite; X3, X4 = NPN transistor, Motorola 2N3904; and X16 = PNP, Power Darlington transistor array, Motorola MJ2500
apart indicated great stability, both of instrument and solutions (Table 111). In Table IV, the "instrumental anisotropy" is evaluated. This factor indicates to what extent the entire detection system is prejudiced in its sensitivity to vertically-polarized vis-a-vis horizontally-polarized light. If no such difference exists, the anisotropy factor is 1. Table V shows a set of data on the excess fluorescence intensity from fluorescein solutions of differing concentraANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
849
IC-II
R33
Figure 6. Current limit circuit (Module C-1)
m -15V
This circuit limits the voltage drop across R40 (Figure 5) and thus limits the lamp current. LED1 goes "on" if the current limit circuit is, at any moment, actually limiting the lamp current instead of its being controlled by the circuit of Figure 5. If LEDl is "on", the voltage from the variable transformer (legend, Figure 5) must be decreased. The parts list includes D1. D2 = Silicon diode, 1N914; LEDl = Light-emitting diode, Monsanto MV5020; P5 = 10KR, Bourns Trimpot 3006P-1-103; R20 = 3.01-KR, 1 %, Corning RN6OC; R21 = 6.8-KR, l o % , Ohmite RC20; R22 = lOO-KR, l o % , Ohmite RC20; R23 = 1-MR, l o % , Ohmite RC20; R24 = 10-KR. 2 % , Corning RL07S; R25 = 2 2 0 4 , 10% Ohmite RC20; R26 = 1-KR, IO%, Ohmite RC20; X2. X5 = PNP transistor, Motorola 2N3906; X6-X10 = Linear transistor array, RCA CA3046; and 22 = Zener diode, 9-V, Motorola 1N4739
+15 V
Figure 7. "Over-Under'' indicating circuit (Module C-ll) This circuit indicates whether or not the variable transformer feeding the lamp power supply (cf. Legend, Figure 5) is set at a point where control will ensue. For this condition, both LED2 and LED3 must be "off". The parts list includes D5-D8 = Silicon diode, 1N914; LED2, LED3 = Light-emitting diode, Monsanto MV2050; P6 = 2-KR Bourns Trimpot 3006P-1-202; R33 = 470-R. l o % , Ohmite RC20; R34 = 4.7-KR. 10%. Ohmite RC20; R35 = 91-n. 5 % , Ohmite RC20; R36. R37 = 2.2-KR, l o % , Ohmite RC20; R38 = 510-R, 5 % , Ohmite RC20; R39 = 220-R. 5%. Ohmite RC20; X11-Xl5 = Linear transistor array, RCA CA3046; and 23 = Zener diode, 9-V, Motorola 1N4739
tion. Linearity of response is shown over a wide range, exM fluorescein. tending down to 8 X The absolute values of the polarizations obtained are difficult to check with certainty because of the lack of materials with accurately known values of p . However, the result obtained for fluorescein a t p H 7 (in SAP) is about 0.0227 (Table 111) a t 19.6 "C. For comparison, a value of 0.017 is given by Pringsheim (17) for disodium fluorescein dissolved in water (pH and temperature unspecified). Polarization measurements can be made with a precision of better than f O . O O 1 unit and intensity measurements well within f0.5%. The sensitivity (as discussed in the introducM fluorescein, Le., tion) of the instrument is about twice the standard deviation in measuring the apparent fluorescent intensity of SAP is equivalent to the excess fluorescence contributed by M fluorescein.
DISCUSSION The underlying goals involved in the design of this instrument were great stability and sensitivity of both intensity and polarization measurements on solutions a t equilibrium. Less importance was assigned to the speed of measurement as would be needed for kinetic studies. To achieve high stability, considerable attention was put into the stabilization (by feedback control) of' the light source and, in addition, the use of a motor driven analyzer rotating a t 90 rpm makes the polarization measurement even more independent of any remaining intensity fluctuations. The lamp regulation shows no discernible warm-up drift, and short- and long-term constancy is better than 0.1%. The use of the photomultiplier only as a photon counter makes the system very insensitive to time-dependent changes in photomultiplier gain. For example, during several hours of warm-up, the gain of our 9635QB may change as much as 20%, while the concomitant change in count 850
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
Figure 8. Signal amplifier and voltage reference generator and amplifier (Modules C-Ill and C-IV) The parts list includes A i , A2 = 471 op. amp., Fairchild U587741393: C1 = 0.1-pf, 25-V, Centralab UK25-104; C2 = 0.01 pf, disc capacitor, Centralab DD103; P1, P4 = lO-KR, Bourns Trimpot 3389P-1-103; P2, P3 = 2-KR, Bourns trimpot 3006P-1-202, front screwdriver adjustment; PT = Vacuum phototube, RCA type 929; R1, R2 = lOO-KR, TI, Corning RL07S;R3, R6 = lO.O-MR, selected RC20; R4 = 1.00-MR. TI, selected Corning RL07S; R5 = 100-KR, TI, selected Corning RL07S; R7 = 1.0-MR, RC20; R8 = lOO-KR, RC20; R9-Rll = 1.00-KR, TI, selected Corning RL07S; R12 = 2.00-KR, TI, selected Corning RLO7S; R13 = lO-R, selected, RC20; R14 = 1.5-KR, carbon, RC20; R15 = 10-KR, 2%. TI, Corning RL07S; R16 = 1-KQ, 2 % , TI, Corning RL07S; R17, R18 = lOO-KR, 2 % , TI, Corning RL07S; R19 = 47-KR. carbon RC20; SW1 = Switch, 2P, 2-6 position, non-shorting, Centralab PA2003; SW2 = Switch, 2P. 2-6 position, shorting, Centraiab PA2002; X1 = Matched dual field-effect transistor, National Semiconductor FM3956; Z1 = 6.4-V Zener, Motorola 1N4560
Table I. Uncorrected p o and IO Measurements"
t , = 0.120 Sample Trial 1 A. I o X PO
B. I o X PO
c. Io x 10-6 PO
Trial 2 A. Io X PO
B. Io X PO
c. Io x 10-6 PO
Excitation:
Lo
0.1442 0.1220 0.1614 0.0151 0.1602 0.1938 0.1424 0.1220 0.1589 0.0153 0.1597 0.1951
Med
t,
t , = 0.140
5
0.160
Hi
Lo
Med
Hi
Lo
Med
Hi
0.4206 0.1211 0.7897 0.0157 0.4720 0.1970
1.629 0.1116 1.836 0.0148 1.448 0.1840
0.1446 0.1180 0.1617 0.0167 0.1597 0.1948
0.4237 0.1159 0.7888 0.0156 0.4711 0.1913
1.629 0.1062 1.831 0.0146 1.452 0.1782
0.4180 0.1212 0.7797 0.0158 0.$704 0.1977
1.618 0.1110 1.805 0.0148 1.443 0.1846
0.1419 0.1180 0.1592 0.0168 0.1597 0.1961
0.4158 0.1159 0.7787 0.0160 0.4695 0.1920
1.618 0.1066 1.806 0.0145 1.418 0.1793
0.4233
1.644
0.1200
0.1101
0.7876 0.0145 0.4733 0.1904
1.824 0.0136 1.449 0.1779
0.1437 0.1236 0.1620 0.0164 0.1600 0.2007
0.4193 0.1199 0.7774 0.0146 0.4717 0.1913
1.617 0.1100 1.806 0.0139 1.444 0.1785
0.1424 0.1235 0.1590 0.0166 0.1595 0.2020
a These data compare observed values of I Oand p o on different samples a t different times and at different levels of illumination. The quantity IO is the average total number of photon counts per second in the direction of observation, while p o is the ratio (DIFF)/(SUM)and is proportional to polarization. The samples were as follows: (A) the fluorescent glass standard, (B) 2 X M fluorescein in SAP, and (C) M fluorescein in 55.5 wt % sucrose. Each sample was measured for nine experimental conditions: three values of gate time a t each of three levels of excitation chosen to give photon counts from 1.4 X lo5 to 1.8 X lo6 sec-'. About 3 X lo7 total counts were obtained for each measurement. Samples were allowed about 15 min to come to constant temperature (19.6 z!= 0.2 "C). Trial 2 was made on the same solutions one month after Trial 1. During the interval, the solutions were stored in the dark at room temperature. Samples B and C were measured in black painted cells.
Table 11. Corrected Measurements of Polarization and Intensitya
t , = 0.120 Sample Trial 1 A. IILIF x 10-6 P B. IILIF x 10-6 P
C . IILIF x 10-6 P Trial 2 A. IILIF x P B. IILIF x P
C . IILIF x 10-6 P
Excitation:
t , = 0.140
t , = 0.160
Lo
Med
Hi
Lo
Med
Hi
Lo
Med
Hi
1.836 0.1785 0.4115 0.0221 0.1620 0.3040
1.832 0.1789 0.4170 0.0223 0.1632 0.3052
1.860 0.1801 0.4188 0.0226 0.1613 0.3064
1.830 0.1792 0.4130 0.0227 0.1618 0.3050
1.821
0.1789 0.4182 0.0228 0.1627 0.3059
1.841 0.1806 0.4218 0.0233 0.1612 0.3068
1.841 0.1793 0.4123 0.0228 0.1615 0.3048
1.835 0.1798 0.4177 0.0223 0.1624 0.3060
1.841 0.1802 0.4206 0.0228 0.1617 0.3062
1.813 0.1782 0.4050 0.0222 0.1615 0.3059
1.814 0.1786 0.4113 0.0223 0.1626 0.3066
1.825 0.1793 0.4138 0.0229 0.1607 0.3072
1.813 0.1790 0.4052 0.0229 0.1613 0.3069
1.809 0.1791 0.4125 0.0229 0.1622 0.3069
1.827 0.1795 0.4136 0.0232 0.1606 0.3078
1.807 0.1793 0.4057 0.0230 0.1615 0.3068
1.799 0.1796 0.4120 0.0230 0.1619 0.3069
1.827 0.1807 0.4140 0.0225 0.1576 0.3073
a These values are based upon the primary data given in Table I and have been corrected by the methods presented in the Appendix.
rate is only 2%. After eight hours of photomultiplier warmup, no subsequent changes in count rate can be found. T h e ultimate sensitivity of a photon-counting apparatus is limited by the background count rate of the photomultiplier, about 100 to 200 sec-l for the 9635QB at ambient temperatures. As pointed out in the introduction, the practical limit a t this time arises from the combined contributions of scattered excitation energy not rejected by the detector filters, residual fluorescence of the cuvette. and
Raman scattering of the excitation energy by water. A typical cuvette containing azide-preserved phosphate buffer a t p H 7 reads lo3 sec-l (when normalized for number of chopper cycles, gate time, and light intensity) in a black-painted cell; this corresponds to the excess intensity contributed by a7X M fluorescein solution. This blank is stable to about 1/2% over time and from cell to cell. The figure of 7 X fluorescein was derived from the data of Table V i n d includes a contribution (-30%) originating from the ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
851
Table 111. Mean and Standard Deviation of Intensity and Polarization Values of Table I1 IILIF x Sample” A. 1st trial 2nd trial B. 1st trial 2nd trial c. 1st trial 2nd trial
P
Mean
U
%
Mean
1.838
0.011
0.6
0.1795 0.0007 0.4
1.815
0.010
0.5
0.1793 0.0007 0.4
0.4168
0.0037
0.9
0.0226 0.0004 1.6
0.4103
0.0039
1.0
0.0228 0.0004 1.5
0.1620
0.0007
0.4
0.3056 0.0009 0.3
0.1611
0.0015
0.9
0.3069 0.0005 0.2
u
%
” For a description of the samples A, B, C, see Table I. Each mean and the intratrial standard error, expressed directly, u, and as a percentage (%) of the measured quantity, is derived from the nine individual values of Table 11.
Table IV. Measured Valuesa of the “Instrumental Anisotropy” at Different Gate Times 7
Gate time, s
PA
6 = -2.54’
6 = -5.570
0.160 0.000189 0.6543 0.6401 0.140 0.00090 0.6921 0.6624 0.120 0.00007 0.6909 0.6445 a These measurements were made on a source of unpolarized light (see Materials) and p = PA17 ideally would have a value of zero. The small “instrumental anisotropy” makes the observed values slightly greater, but the effect may be completely neglected except in cases where the absolute value of p is needed to the greatest accuracy. The values of 6, a constant phase angle, also ideally zero, were determined by optimizing the least squares fit of data from varying gate times. T w o values of 6 are quoted since the values of 6 = -2.54” was found to fit best the data from the glass standard and fluorescein in SAP, while 6 = -5.57’ was best suited to the data for fluorescein in 55.5 wt % sucrose. This difference in 6 is produced by the optical rotation of sucrose. The parameter 7 converts the readout of the computer to polarization values. A full discussion of these parameters is given in the Appendix.
photomultiplier dark current. Since the latter quantity is nearly constant and can be determined with high accuracy, it can be justifiably deducted in computing the “fluorescein equivalent” of the blank, which in this case would then be 5 X M. This latter procedure has the added advantage that the result is independent of incident light intensity. In systems where serum or serum proteins are present, the magnitude of the blank may be many times this figure. In these situations, the attainable sensitivity of measurements will be limited by the precision with which the protein or serum blank itself can be measured. As higher count rates are encountered, coincident photon events (i.e., those occurring within the effective dead time of the entire detecting system) cause nonlinearity of response. The ability to reduce the incident intensity by 500 to 1affords a corresponding increase in the linear operating range of the instrument. At a count rate of lo6 sec-l, the correction is 7.6%. While the periodic nature of the in852
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
Table V. Excess Fluorescence Intensity as a Function of Fluorescein Concentration c. molA. ASM ASM/c 1.6 x 103 2.0 1.64 x 104 2.05 5.00 X 1.005 x 105 2.01 8.00 X 1.63 x 105 2.038 1.40 x 10-lo 2.815 x lo5 2.01 2.00 x 10-10 4.05 x 105 2.02~ Fluorescein was diluted volumetrically in SAP (see Materials) to the concentrations indicated and the blank and samples were counted for 200 chopper cycles, or 66% s, total elapsed time, with a gate time, t,, of 150 ms, and a lamp intensity factor of 20. The actual time during which counts were being accumulated by the computer was 60 s = 2Nt,. The differences ASM between sample and blank were calculated for each concentration as shown. A least squares treatment showed that the data fit the equation ASM = 2.021 X lOI5c 29, with a regression coefficient of 1.000. The standard errors of estimate The magnitude of are: U A S M . ~ = 893 and ~,.;\sM = 4.42 X SM for the blank in this set of measurements was 1.40 X lo4, of which about 4.3 X lo3 was due t o photomultiplier dark current. 8.00 x 10-13 8.00 x 10-l2
+
tensity as seen by the photomultiplier complicates corrections, the Appendix describes a method to obtain consistent corrections for polarization and intensity to count rates of 2.5 X lo6 sec-l. At a lamp intensity factor of 1, this count rate corresponds to a fluorescein concentration of 1.2 X M. The insertion of a neutral density filter could further extend the range of the instrument, but selfquenching becomes a problem above about M. Thus, the useful range of the instrument extends from about 5 X M to M fluorescein. Errors in measuring intensities and polarizations can arise from several other sources. The short-term precision of measurement is influenced by the Poisson character of the photon-counting technique which introduces large random fluctuations a t low count values. If the expected SUM count is n , the variance of both the SUM and the DIFF is also n. Thus, while the standard deviation (4) may be small relative to the intensity (SUM), it will be larger relative to the polarization as DIFF may be only several percent of the SUM a t low polarizations. This source of noise may be reduced as much as desired by increasing either the observation time or the excitation intensity. Estimates of the variance due to counting statistics are presented in the Appendix. However, the ability to increase the count by increasing the time of observation has the advantage of not exposing the sample to very high excitation intensities. Even fluorescein itself shows some sensitivity to photochemical decomposition. The fluorescence of a M solution of fluorescein in phosphate-buffered saline, exposed to average laboratory ambient illumination, has a half-life of about two weeks. The identical solution, stored in the dark, shows less than 2% decay over a period of a month. Because of the temperature dependence of viscosity, the precision of temperature control has a direct influence on the observed polarization. The cell holder on this instrument is regulated to about f0.05 O C , but there is a thermal gradient of over a degree from the bottom of the holder to the top, Thus, as much as 0.2 “C fluctuation may occur in the illuminated area of the cell. One would therefore expect systematic errors of h0.005 unit in reading the polarization of fluorescein in such a sucrose solution. Actual measurements show short-term fluctuations of fO.001 unit and long-term reproducibility (over several weeks) of f0.002
Z
be accumulated in one of the two high-speed registers in the SSR 1110, for a specified chopper gate time interval, t,. Negative transitions cause accumulation in the other register. The chopper gate time is always taken to be less than a/2w to ensure that each register accumulates counts only during its own chopper state. The gate time is controlled by a manual adjustment on the SSR 1110 computer and is set at present a t 150 ms. Fluorescence is observed, by photomultiplier P M on the y axis, to have a component exciting u photoelectrons per second when the analyzer A is oriented parallel to the z axis, and h photoelectrons per second when A is oriented perpendicular to the z axis. The expected rate of photoemission at time t in the direction of observation is then Z ( t ) = u cos2 ( u t
+ 6) + h sin2 ( w t + 6)
(ii)
The time interval beginning when the chopper signal goes “on” and ending after the chopper gate time, t,, has elapsed, is denoted JA. We assume the analyzer is perfectly symmetrical and so consider only one chopper transition of each type. Thus,
Figure 9. Rotating analyzer
unit, for a 10-8 M solution of fluorescein in phosphatebuffered 55.5 wt. % sucrose. The polarization of this solution is about 0.306. By contrast, the polarization of a glass standard shows short-term stability of f O . O O 1 unit, and long-term reproducibility of better than f0.0005 unit, a t a polarization of 0.180. The 1-digit uncertainty inherent in the digital display is the limiting short-term error factor a t high count values. This ranges from 1%for a display of 100 to 0.1% for a display of 999. The data in Table V shows fluorescence intensities for down to 8 X fluorescein solutions ranging from 2 X M. The linearity of these data is dependent not only upon a linear instrument response, but also upon the accuracy of dilution and, at low concentrations, reproducibility of the blank. The constancy of the quantity ASMIc indicates that none of the above factors is introducing appreciable errors in the measurements and also that the procedure of deducting the reading for the blank from that of the sample to obtain an “excess fluorescence” is valid over a wide concentration range.
APPENDIX I. Computing Polarization with the Rotating Analyzer. Suppose that light, from a point on the x axis and with polarization parallel to the z axis, traverses the sample at the origin 0 (Figure 9). An analyzer A located along the y axis rotates with angular velocity w in a plane perpendicular to the y axis. At time, t , the angular orientation of the transmission axis of A is
Similarly, the time interval beginning when the chopper signal goes “off’ and ending when the chopper gate time has elapsed is denoted J g n - a
J - --+t,
- (*d 4w
Note that these time intervals correspond to the respective 0 intervals (-a14 6, -a14 6 ut,) and (n-14 6, n-14 6 ut,). The measure of each interval is the chopper gate time t,. Define
+
where t is chosen so that a t t = 0, 0 = 6. The rotation of A causes the attached chopper (shown as alternate light and dark areas near the periphery of A) to alternately transmit (“on”) or not transmit (“off”) light passing from the LED to the LST. The transitions occur a t four equidistant values of 0, the positive going (“off)’ to “on”) transitions occur a t the orientations (0 = -x/4 + 6) and (0 = 3x/4 6), while the negative going chopper transitions occur at (0 = s/4 + 6) and (0 = 5*/4 6). The angle 6 is a measure of the alignment of the transmission axis and the chopper, and the position of the light beam between the LED and the LST. If both the transmission axis of A and the light beam simultaneously pass through the center of one of the light areas of the chopper, then 6 = 0. In the present instrument, 6 is -2.5’ and w is 3r radianslsec. Each positive transition of the chopper causes subsequent photoelectron events to
+
+
+ +
+
+
and (vi) the average value of I ( t ) over the intervals J A and J g , respectively. Assuming w is constant, integration reveals
(+) + h (?) B = u (+) +h
A= u
(y)
(vii) (viii)
where
6)
0=wt+6
+
(iii)
1=
sin ( u t , + 26) sin (ut,) ut,
Thus
A+B=u+h
A -B =
T(U
-h)
(X)
(xi)
Since the above reasoning applies to measurements both on the blank and on the sample, we may write
AA AA
+ AB = Au + Ah
-
~(Au Ah)
(xii) (xiii)
Since p is defined as AV - Ah
p=zzTi ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
(xiv) 853
SM
it may be found from
(xxiii)
and The expression of p in Equation xv in terms of counts’per unit time is equivalent to the more usual (Equation xiv) definition of p in terms of intensities if only a single wavelength is considered. Over the small range of wavelengths in question, we feel that any discrepancies introduced by the wavelength dependence of photon energy would be unimportant. The idealized analysis of Section I neglects several important practical limitations which are dealt with in the following three sections. 11. Correction f o r Dead Time. The number of photoelectrons emitted in any time interval is a random variable X having the Poisson distribution
P (X= ni = e-X-
IO
1
- 710
(xvii)
The accumulating registers of the SSR 1110 are denoted DATA and BKG. For a given measurement, each sums the photoelectrons counted during its respective time interval JAor J B , for as many intervals N as are selected by the operator. When a measurement is terminated, the contents of these registers are converted to scientific notation and displayed with three significant digits, plus an exponent ranging from 2 to 7 . The sum and the differences of these registers are also computed and displayed, in the same format, as the quantities SUM and DIFF. We choose the symbols DT, BK, SM, and DF to denote the contents of the respective displays DATA, BKD, SUM, and DIFF for a given measurement. These symbols will be used contextually to represent either their realized value for a measurement, or their statistically expected value. The most simple method of correction would be to apply Equation xvii to DT and BK, separately. However, this method of correction does not utilize the capability of the computer to give also both SM and DF to the same number of digits as it does DT and BK. We have D T = Nt,Ao
(xviii)
BK = Nt,Bo
(xix)
SM = Nt,(Ao
+ Bo)
(xx)
DF = Nt,(Ao
- Bo)
(xxi)
where Ao, Bo are the observed (or “counted”) photoelectron rates corresponding to A, B. The average observed rate I Ois
= (Ao
+ Bo)/Z
2Ntg’ A simple method for the calculation of p was described under “Methods”. An alternative, more complicated, procedure which, however, utilizes the full three-digit capability of the computer to obtain values of DF and SM directly employs the function p , where po = DF/SM. The value p corrected for dead time is given by P =
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
~
PO
7SM(1 - po2) 12Ntg
(xxv)
For the calculation of p by this method, first obtain values of both po and IO for both sample and blank. Correct these quantities for dead time by Equations xxv and xvii, respectively. Compute p from the equation
’=,(
IP I --I IBPB) B
(xxvi)
where the subscript B indicates a quantity pertaining to the blank. 111. The Statistical N a t u r e of the Measurements. The statistical nature of photoelectron emission imposes random variations on the number of pulses per unit time received by the amplifier/discriminator. At low count rates, as are encountered in the measurement of the blank, this variation may be quite large and can be minimized by selecting large counting periods. When the variations resulting from physically reproducing a duplicate solution or blank predominate over those arising from photoelectron statistics, then increasing the counting interval further is of no benefit. When the counting rates are high, with values of S M and DF over lo4, it is appropriate to approximate the distributions of these quantities by the normal distributions having the common variance SM and having respective means SM and DF. This leads to a standard result
--- -a u(SM) SM
1
(xxvii)
A second important relationship in terms of p is derived below: (xxviii) We suppose A, B are independent random variables having first moments A, B and variances A, from their Poisson distribution densities:
For large A, B, we approximate these densities by the appropriate normal densities
(xxii)
Combining Equations xvii through xx, we obtain 854
\A
An
(xvi) n! where X is the expected number of photoelectrons in the time interval. The random variable X has mean and variance A. On the interval J A , X takes the value At,. On the other hand, the amplifier/discriminator has a finite resolution time 7. Some photoelectrons, those occurring within the time 7 after a counted photoelectron, are not counted. Thus, the observed counting rate I O is systematically less than the true rate, I . A first order correction for this effect is given by (SSRManual on Model 1120 amplifier/discriminator) the relation
I=-
(xxiv)
respectively.
Define r = B / A ; with density
B/A, then r is a random variable
3
w(r) =
i-
fB(rt)tfA(t)dt
-
We shall approximate iu by using the densities f and g:
Lrn
w(r) z
g(rt)tf(t)dt
which, after substitutions and rearrangement, yields
where k
2
-. F~A r2
+ F’
h*(r
01
+ 1).
Now we observe that the exponential term under the integral is just m
t
n
t-CY (T)
(where n ( r ) is the normal density) and that, furthermore, for large A, the integrated probability density is vanishingly small for nonpositive values o f t . Hence,
r
te-1/2(t-a)2/kZdt
-
where d(p) is the half-width of the confidence interval for p . Let q be 1 - l / e (about 0.632). As A a,m(F,q) 1for all values of F and q . Taking m ( f , q )to be 1, we have as the result, Equation xxviii. IV. Instrumental “Anisotropy”. Slight imperfections or inhomogeneities in various elements of the optical system may induce systematic differences such that a sample actually having zero polarization shows a value of p = PA. One of the chief sources for such an “instrumental anisotropy” lies in the often-observed dependence of photocathode sensitivity upon azimuth of polarization. Once the value of P A is known, it may be used to correct experimental measurements by the equation
b
2
P
- (1 - P 2 ) P A
(xxix)
where i, is the value of p corrected for the factor PA. This factor for a given instrument configuration is a function of gate time t,. Thus, measurements made a t a constant gate time need not be corrected unless the absolute polarization is needed.
te-l/2(t-a)2/k2dt =
TABLE OF SYMBOLS
and
d = angular orientation of rotating analyzer, B = ot
+
6 = angular velocity of rotating analyzer 6 = phase lead of the analyzer relative to the chopper t = time t , = the chopper gate time interval J A , J g = time intervals, selected jointly by the rotation of the chopper and by the selected chopper gate time t , r = response time of the amplifier/discriminator u = photoelectrons per second of the vertically polarized fluorescence energy reaching the photocathode h = photoelectrons per second of the horizontally polarized fluorescence energy reaching the photocathode I ( t ) = counts per second at time t of the fluorescence energy reaching the photocathode A, B = the average counting rate of fluorescence energy reaching the photocathode during the respective time intervals J A .JH. A = the value of a particular quantity for the sample minus that for the blank 0 = subscript denoting “observed” po = a measure of polarization defined by
(L:
We make the further substitution using
u 3 1 and d r = Fdu T
to obtain
-
The density u decreases as 1/u2as u m , hence is related t o the Cauchy distribution. I t has no moment of one or greater, that is, neither the average nor the variance exist. However, as A a,
-
Thus, u is “almost” normally distributed with mean value 1 and variance (1 l/F)/L.T h a t is, r is almost normal with mean F‘ and variance F(F l ) / L . A confidence interval containing the observed value of u with probability q is “almost” the same size as the confidence interval containing a normally-distributed random variable, having mean value 1 and variance (1 lh)/A, with probability q , for q < 1. Now
+
+
+
ap = - ___ 2 p = - - -1- - F and 1
+
~ aF
Assigning the confidence interval of half-width d(r) m(i=,q)sto r where s2
=
T ( F i- 1) ~
A and rn(r;q) is a function of have
and confidence level q , we
p = 9=
the corrected value of po the ratio of p/p, found to be sin ( u t g 26) sin(ot,)
+
9=
utg
N = the number of chopper cycles per measurement DT = the number of events counted by the DATA register of the SSR 1110, being Nt,Ao BK = the number of events counted by the BKG register of the SSR 1110,being Nt,B,l SM = the number of events displayed in the SUM register, being D T BK DF = the number of events displayed in the DIFF register, being D T - BK I o = the observed number of photon counts per unit time found from SUM Io = 2Ntg I = corrected value of photon counts per unit time
+
I=-
Io (1 - l o r )
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
855
P = polarization, d e f i n e d by p = -AV - Ah Av Ah
+
LITERATURE CITED (1) W. 8. Dandliker and V. A. deSaussure, Immunochemistry, 7, 799 (1970). (2) W. B. Dandiiker. R. J. Kelly, J. Dandliker, J. Farquhar, and J. Levin, Immunochemistry, I O , 219 (1973). (3) W. B. Dandliker, H. C. Schapiro, R. Alonso, and D. E. Williamson, San Diego Symp. Biomed. Eng., 3, 127 (1963). (4) L. Monnerie and J. Neel, J. Chim. fhys., 62, 504 (1965). (5) D.Deranleau, Anal. Biochem., 16, 438 (1966). (6) F. Aurich and E. Lippert, Spectrochim. Acta, 22, 1073 (1966). (7) G. Weber and B. Bablouzian, J. Bo/.Chem., 241, 2558 (1966). (8) 8. Witholt and L. Brand, Rev. Sci. Instrum., 39, 1271 (1968). (9) R. H. McKay, Arch. Biochem. Biophys.. 135, 218 (1969). (10) C. Rosen, Acta Chem. Scad., 24, 1849 (1970).
(1 1) S. Claesson and H. Odani, Discuss. Faraday Soc., 49, 268 (1970). (12) J. Lavorel, C. Vernotte, B. Arrio. and F. Rodier, Biochimie, 54, 161 (1972). (13) R. D. Spencer, F. B. Toledo, B. T. Williams and N. L. Yoss, Clin. Chem. ( Winston-Salem, N.C.), 19, 838 (1973). (14) R. E. Curry, H. L. Pardue, G. E. Mieling, and R. E. Santini, Clin. Chem. ( Winston-Salem, N.C.), 19, 1259 (1973). (15) J. E. Wampler and R. J. De Sa, Anal. Chem., 46, 563 (1974). (16) W. B. Dandliker, H. C. Schapiro, J. W. Meduski, R. Alonso, G. A. Feigen, and J. R. Hamrick, Immunochemistry, 1, 165 (1964). (17) P. Pringsheim, "Fluorescence and Phosphorescence," Interscience, New York, 1949, Table 69.
RECEIVEDfor review August 21, 1975. Accepted February 9, 1976. Supported by Cordis Corporation, Miami, Fla., Research Grant No. GB-31611 from the National Science Foundation, and Contract No. N01-CB-43905 from the National Cancer Institute.
Exchange Kinetics at Ion-Selective Membrane Electrodes Karl Cammann' and G. A. Rechnitz* Department of Chemistry, State University of N e w York, Buffalo, N e w York 74274
Mechanlstlc information concerning the operation and selectivity of several ion-selective membrane electrodes Is obtained from exchange current measurements carried out under varying solution conditions. Particularly high exchange current densities were found for AgZS-based membrane electrodes. The potentiometric behavior of LaF3 type F--selective electrodes in the presence of OH- or La3+ can be successfully explained on the basis of the exchange current density estimates obtained in this study.
This study attempts to examine the kinetics of chargetransfer processes occurring a t phase boundaries of ion-selective membrane electrodes in order to test whether or not thermodynamic equilibrium is established a t these boundaries. Traditionally, the heterogeneous kinetics of charge-transfer reactions are described electrochemically in terms of the standard exchange current density, ioo ( I ) . For metal electrodes, the determination of exchange current densities from current-voltage curves is not difficult, but the determination and evaluation of quasi-stationary current-voltage curves for the phases of ion-selective membrane electrodes involves severe experimental and interpretive difficulties. In the case of metallic electrodes, both the chemical composition and electrostatic potential remain constant in the metal phase and the potential difference between the electrode phase and the solution arises mainly between the metal phase and the outer Helmholtz plane. This is not so in the case of ion-selective membrane electrodes which have high ohmic resistances. Since the precise details of the chemical and electrostatic potential distribution across ion-selective membrane electrodes are not known, an exact theoretical treatment of electrode kinetics is not attempted here; instead, the concepts and methods of electrode kinetics developed for metal electrodes are employed as a first approximation and are used to estimate exchange current densities which should be viewed strictly as apparent exchange current V i s i t i n g Researcher
856
from t h e U n i v e r s i t y of M u n i c h , Germany.
ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976
densities, i&. With this qualification in mind, the methods employed in this study are based on the Butler-Volmer equation, which describes the stationary current-voltage curve, e.g., for one redox reaction (with 2 = 0) a t a metal electrode ( 2 , 3 )
where i is the current density a t overpotential 7, io is the exchange current density a t 7 = 0, and p is the transfer coefficient. Since the absolute potential difference between two phases is not measurable, one should compare individual electrode reactions according to their kinetics a t 7 = 0. For ion-selective membrane electrodes the exchange current density also is more characteristic of electrode kinetics a t the equilibrium potential than the absolute heterogeneous rate constant a t zero field strength. Jaenicke and Haase ( 4 ) have already employed principles of metal-electrode kinetics for the dissolution of '!salt electrodes" consisting of thin (-10 bm) silver halide layers on silver metal and developed individual expressions for the exchange current densities of cations and anions, respectively. In case of ion electrodes, of course, the exchange current density of the counterion can usually be neglected by proper selection of the electrolyte. There are basically two methods for determination of the exchange current density from equation 1. A t low overpotentials ((10 mV) equation 1 reduces t o
.
F =io-? RT and the transfer coefficient drops out. Assuming the validity of the simplified equivalent circuit shown in Figure 1,we can define a charge-transfer resistance Rt, for the behavior of a phase boundary under current flow, as 1
(3) with knowledge of Rt we can then calculate io from io = RT/FRt
(4)
