Anal. Chem. 1984, 56, 1401-1403
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Multipath Cells for Extending Dynamic Range of Optical Absorbance Measurements Purnendu K. Dasgupta Department of Chemistry, Texas Tech University, Box 4260,Lubbock, Texas 79409
A reflective helical cell Is described for extending the dynamic range of optical absorbance measurements. This cell behaves as a multipath cell. The effective path length of a multipath cell varies with solution absorbance and Is substantially greater for solutions of low absorbance than for solutions of high absorbance, thus extending the dynamic range of possible measurements.
Apparently, the advantage of multipath cells for making optical absorbance measurements for analytical applications has never been documented. Recently, Lei et al. (I) have reported the use of a long path capillary cell for optical absorbance measurements of solutions of high transmittance. This cell behaves as a multipath cell as opposed to a multipass cell commonly used with gases which involves multiple reflections of a single beam traversing the cell, after the original design of White (2). The work of Lei et al. was designed, however, for exploiting very long path lengths rather than the effect of multiple path lengths. I have carried out investigations with an internally reflective helical absorption cell which is small enough to fit in sample compartments of conventional spectrophotometers. This cell behaves as a multipath cell due to the nature of multiple reflections of a finite size beam on a concave reflecting surface. The effective path length of a multipath cell varies with the absorbance of the solution. Because the effective path length is much greater for solutions of lower absorbance compared to solutions of higher absorbance, the dynamic range of measurements is greatly extended. Hansen has previously described (3) a helical cell which involves multiple attenuated total reflection; this cell was designed to attain very small path lengths.
THEORY Consider a beam of total power Powatts entering a cell system consisting of i separate cells of path length b; cm each, arranged in parallel. Let us assume that a beam splitter splits this beam without loss into individual fractions ji,which then proceeds through the individual cells. See Figure 1 for a schematic representation for a system with i = 4. Thus, Po = PoCf,where Cf; = 1. If all the cells are filled with the same solution of c molar concentration and of molar absorptivity e, the absorbance, A,, in cell i is given by Ai = tb,c (1) the transmitted power, P,, through cell i is then calculated from the transmittance T, Ti = Pi/fiPo = lo-*' (2) to be
Pi = fiPo(lo-fbq (3) If the transmitted beams are now recombined before measurement, the total transmitted power is given by P = CP, = PoC(fi(lO-'*lc)) (4) The effective absorbance of the entire cell system is then given by 0003-2700/84/0356-1401$01.50/0
A = -log P/Po = -log (E(f,(lO-fb~c)))
(5)
The effective path length, 6, can be defined as
6 = A/tc
(6)
Equations 5 and 6 combine to
b = -(tc)-l
log ( E ( f t ( l O - W )
(7)
If comparable j , values are assumed, consideration of (7) indicates that for large values of cb,, such individual terms will contribute little to the summation term in (7) and thence have little effect on 6. In other words, for a given value of e, larger values of b, are only important if c is small. Consider for example, a combination of cells of path lengths b, = 1, 2, 5, and 10 cm, respectively, each receiving 25% of the total incident power (fl = fz = f3 = f4 = 0.25). The transmitted light is combined and registered on a detector. It is easily computed from (7) that the effective path length is 4.50,4.49,4.36,3.37,1.56,and 1.30 cm for Al, (conventional absorbance in 1cm cell) values of 0.OOO1,0.001,0.01,0.10,1.0, and 2.0, respectively (e.g., for Alcm = 1.0, 6 = -(l)-' log (0.25 0.25 X lo4 0.25 X = 1.56). Even x 10-1+ 0.25 X if j, values are quite different from each other, the nonlinear dependence of the effective path length on solution absorbance is maintained. Consider now (7) for very small values of eb,c (solutions of high transmittance). For very small values of x , 10' 1+ x In 10 and thus (7) becomes
+
+
6 = -(t~)-' log (E(f,(l- 2.303tb,~)))
(8)
Recognizing that Cf, = 1 and converting to natural logarithms, (8) becomes
6 = -(2.303~c)-~In (1- 2.303C(f,tbic)) Since for small values of x, In (1 - x ) himiting
= C(fcbJ
N
(9)
-x, (9) becomes
(10)
This limiting effective path length is 4.50 cm in the numerical example given above and lies in an absorbance domain likely to be of experimental interest. The limiting amplification for solutions of very low transmittance is simply controlled by the shortest path length. Attainment of this lower limit may or may not be important because the absorbance domain at which this limit is approached may well be too high to be of practical interest. It is clear that such a nonlinear dependence of the effective path length of the cell system upon the solution absorbance greatly extends the attainable dyaamic range of measurements. Taking the above numbers as a practical example, no commercial instrument will cover a range of 0.00045-9.00 absorbance units (a single conventional cell of 4.50 cm path length), while many are capable of covering an absorbance range of 0.00045-2.60 (for the multiple cell system given above). It is physically inconvenient to deal with a multitude of cells of different path lengths along with beam splitting and recombination necessary for practical instruments. For this reason, an internally reflecting tubular helical cell was con@ 1984 American Chemical Soclety
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
Table I. Power of Transmitted Light as a Function of Sample Absorbance sample abs in conventional 1-cm cell
wb3
J43.
,J ..... c
Zfi'i
b4
f4P,10-'b4' J.......
-
. . . ~ .............
Cells
Figure 1. A hypothetical multipath cell experiment with four cells of different path lengths. All cells contain the same solution of c molar concentration with a molar absorptivity of e. IS
transmitted transmittance, % power, pW
absorbance
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0075 0.0100 0.0150 0.0200
1015 100.oa O.OOoa 896 88.3 0.054 814 80.2 0.096 754 74.3 0.129 705 69.5 0.158 671 66.1 0.180 610 60.1 0.221 571 56.2 0.250 491 48.2 0.315 443 43.6 0.360 a The transmittance is arbitrarily defined as 100% with pure water in the cell. r
1
C
Flgure 2. Experimental setup: L, laser; F, Interference filter; C, cell; D, photodiode; A, to sample solution; B, to suction pump; W, washer: S, black cardboard shield.
ceived. When a spot of finite diameter is incident on a concave reflecting surface where the spot diameter is not negligible compared to the radius of curvature of the reflecting surface, the reflected image is magnified and the divergence increases continually with each succeeding reflection. The divergence is a direct measure of the distribution of the available path lengths. Thus the net result is that some portions of the transmitted light have traversed significantly longer paths compared to other portions; i.e., the cell behaves as a multipath cell.
EXPERIMENTAL SECTION A three-turn coil provided with inlet and outlet connections (1mm i.d.1 was made from a 3 mm i.d. Pyrex tubing with a coil diameter slightly less than 1cm. The cell was silvered heavily internally by classical procedures (4) to provide a reflecting swface. Black metal washers of appropriate dimension were cemented to the open ends of the tube to prevent the glass body of the cell from acting as a light conduit. Thin optically flat cover glasses (as used in optical microscopy)were cemented on the open ends of the cell next, to provide the optical windows. The optical experiments were carried out with an argon-ion laser as the light source (Model 162-03,Spectra-Physics,Mountain View, CA) and a photodiode detector (photometer/radiometer Model 301, Photo Research, Burbank, CA). The 514.5-nm line from the laser was isolated with an appropriate interference filter and allowed to be perpendicularly incident on the cell window without further focusing. The total incident power was 1.10 mW. The photodetector was shielded from extraneous light by a black cardboard and placed close to the exit cell window. The experimental arrangement is shown in Figure 2. The cell was initially filled with membrane fiitered (0.1 pm) distilled deionized water, with appropriate care taken to exclude air bubbles. Once the cell was placed in the optical setup, further refillings were carried out by sucking the new solution in with a peristaltic pump with the cell inlet being connected to the sample solution with a PTFE tubing. This prevented possible errors due to physical movement of the cell between experiments. At least 100 cell volumes (approximately 70 mL) were allowed to pass through the cell for each new sample, to ensure that the contents of the cell were representative of the new sample. Solutions of standard absorbance were made from Beryllon 11, a dye that displays strong broad-band absorption around 515 nm. The reported absorbance values in a 1-cm cell (with water set to zero absorbance in the same cell) as given for the various solutions
id
104
-005
Absorbance
-01
.015
in
1 em
.Ob
1
Cell, *lcm
Figure 3. Dependence of the effective path length of the reflective hellcal cell upon solution absorbance.
were obtained with a Cary Model 219 spectrophotometer(Varian Associates, Palo Alto, CA) with a 0.5 nm bandwidth. Due to pulsations caused by the samplingpump, the noise is greater while the pump is operating. Therefore, the reported results represent readings taken within the first 2 min of stopping the sampling flow. If the cell is left in static mode, the observed attenuation increases slightly with time, presumably due to heating and consequent generation of thermal gradients within the cell. The detector reading was set to zero initially with the laser on and with a very concentrated solution of the dye in the cell. The absolute value of the detector reading under this condition was 50 pW above the detector reading in total darkness (laser off) and resulted primarily from ambient reflections. All experiments were conducted in a dark room 'at ambient temperature.
RESULTS AND DISCUSSION The results are reported in Table I. The effective path length of the cell is shown in Figure 3 as a function of the conventional absorbance of the solution in a 1 cm cell (Alcm). Compared to the path length of 10 cm (the length of tubing from which the helix was constructed), the effective path length was greater by a factor of 5.4 at the lowest absorbance studied (Alcm= 0.0010). This may be compared to the factor of 3 increase in effective path length observed by Lei et al. for their 1 m long capillary cell at Alcm = 0.0003. We can directly compare the increase in effective path lengths at Al, = 0.006, which Lei et al. in their system found to be 1.3, while in our system it is 3.3 (Figure 3). Further, it is much easier to work with the design reported in this paper compared to a 1 m long linear cell. While the cell of Lei et al. behaves as a multipath cell also due to divergence of the incident beam, the considerations are somewhat different for the two systems. The cell of Lei et al. is essentially a straight glass capillary tube with a reflecting surface (aluminum tape or silver mirror) provided on the outer wall of the glass tube. The multipath effect was presumed to be due to the various paths originating from: (a)
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
direct linear transmission of light (as in conventional cells), (b) reflection of light at the inner cell wall, and (c) reflection of the light at the outer cell wall. If a nondivergent source, e.g., a laser, is used, the fraction of the incident power traveling through (b) and (c) is very small for a perpendicularlyincident beam and the nonlinear dependence of the effective path length upon solution absorbance is also very small. Even with a divergent source, the bulk of the light travels through (a) and the relative nonlinearity is significantly smaller than that in our system. Clearly, the primary objective of the study of Lei et al. was not centered on exploiting the multipath effect, nor do the authors emphasize its advantages. The effective path length is acutely dependent on the angle of incidence of the beam in their system, and this has been considered in some detail. Preliminary experiments indicate that the relative dependence of the effective path length in our system on the angle of incidence is smaller. There is of course, no possible means for light to be transmitted through the helical cell without undergoing any reflections. The degree of the nonlinear dependence of the effective path length on solution absorbance is also related to the initial divergence of the incident beam; indeed a slight decrease in the extent of nonlinearity has been observed when the incident spot diameter is reduced by focusing with auxiliary optics. It is interesting to note that an internally silvered linear cell of ellipsoid cross section has been used to obtain very long path lengths (5). In this work, the incident laser beam struck the reflecting surface tangentially, thus producing a very long path length by specular reflection. A significant shortcoming of the present cell needs to be mentioned. It is obvious from the presented data that the overall transmittance of the cell used in this work is poor. Even with only water in the cell, the cell attenuates the incident light by nearly 6 orders of magnitude (9.22 X lo-'). This is caused by the poor reflective properties of the silver surface on the cell wall. The conditions employed for silvering resulted in a relatively thick silver layer; the silver surface on the glass side displayed good reflecting properties while the reflecting properties of the exposed silver surface was relatively poor. With optically flat glass substrates (thickness 0.14 mm) silvered in a similar fashion, we have observed (using the same source and detector) that the reflectivity of the exposed silver surface is 92.6% while the glass surface is 95.8% reflective. Under identical conditions, specially coated mirrors for use in laser optics (Spectra-Physics) showed 98.1% reflectivity. Coatings of platinum or gold deposited on the surface by oxidative thermal decomposition of a suitable organometallic precursor (6) reportedly produce a much better reflecting surface on the exposed metallic side. Work is in progress to determine the improvement in transmittance with such cells. An alternative approach involving silvering the outer wall is being pursued with helices made from very thin-walled NMR tubes; preliminary experimentsindicate a 2 order of magnitude improvement in transmittance for otherwise identical cell dimensions. However, source powers of the order of 1 mW are still desirable for measuring high absorbance values in
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order to maintain an acceptable signal/noise ratio. The optimization of cell geometry and the angle of incidence for maximal exploitation of the nonlinear behavior is in progress. The maximum effective path length of the cell is dependent on the diameter and the pitch of the helix and the angle of incidence. Consider that at the lowest absorbance studied (Alcm= O.OOlO), the effective path length of the cell is 54 cm. To attain the same effective path length in a linear 10-cm cell with reflective walls, the angle of incidence needs to be 79.33O (sec 0 = 5.4); the beam will then undergo 176 reflections through the 3 mm diameter, 10 cm long tube, traversing 3.07 mm between each reflection. With a surface reflectivity of 92.6%, the observed attenuation of the helical cell, 9.22 X lo-', is accounted for by 181 reflections, which correlates well with the 176 necessary reflections calculated above. If the reflectivity is improved to 95.8%, as noted earlier for the glass side of the mirror, 176 reflections will lead to an attenuation of only 5.25 x (200 reflections will lead to an attenuation of 1.88 X The multipath effect is expected also for a linear cell of circular cross section (linear tube) inasmuch as the reflecting surface is concave here as well, regardless of whether the cell is externally or internally silvered. For internally silvered cells, the possibility of separate reflections at the glass surface and at the mirror surface does not exist and cannot become a contributing factor toward beam divergence and the consequent multipath effect. However, achieving very long effective path lengths with a linear cell, without greatly increasing the linear dimension requires an angle of incidence that will be difficult to attain with entry and exit windows placed in the conventional position, i.e., perpendicular to the long axis. With a beam entrance and exit system based on small windows created on the perimeter of the tube, it will be difficult to avoid the contributions of the cell wall acting as a light conduit. It is concluded therefore that the helical cell is preferable if it is desirable to keep the total cell volume small. The fact that the effective path length of the helical cell is less dependent on the angle of incidence is also a factor in favor of choosing it over the linear geometry. While a detailed theoretical rationale for this phenomenon cannot be provided at the present time, it is believed that the curvature of the reflecting surface in the third dimension plays an important role.
ACKNOWLEDGMENT I thank Victor L. Johnson for his expert workmanship with glass and Kevin Jordan for experimental assistance.
LITERATURE CITED (1) Lei, W.; Fujiwara, K.; Fuwa, K. Anal. Chem. 1983, 55, 951-955. (2) White, J. U. J. Opt. SOC. Am. 1942, 32, 285-288. (3) Hansen, W. N. Anal. Chem. 1983, 35, 765-766. (4) Scott, R. B.; Cook, J. W.; Brickwedde, F. G. J. Res. Nafl. Bur. Stand. (U.S.) 1931, 7, 935-943. (5) Walker, D. C. Can. J. Chem. 1967, 45, 807-811. (6) Engelhard Industries Dlvlsion, Engelhard Corp., East Newark, NJ.
RECEIVED for review November 4,1983. Accepted March 5, 1984.
