Anal. Chem. 1981, 53, 1373-1376
1373
Effect of Detector Nonlinearity on the Height, Area, Width, and Moments of Peaks in Liquid Chromatography with Absorbance Detectors Lynda M. McDowell, Willlam E. Barber, and Peter W. Carr” Department of Chemistry, Smith & Kolthoff Halls, Unlversity of Minnesota, Minneapolis, Minnesota 55455
The effect of nonlinear absorbance detectors on chromatographic peak height, area, centroid, and second moment is experlmentally investigated. In accord with theory developed previously, those properties related to the even moments, such as the area and peak width, are most strongly influenced by detector nonllnearity, whereas odd moments of symmetrlc (Gaussian) peaks are niot altered. Phis is in contrast to the behavior of the effect of nonlinear isotherms which shift the positlon of the peak ceatrold as well as alter peak symmetry. We have shown that evlen with relatively narrow spectral slit widths (2 nm) absorbance detectors can be quite nonlinear even at low concentration (1 mM) and at absorbances considerably below the StriBy radlation llmlts of the Instrument.
In a previous paper ( I ) we have shown that nonlinear detectors can have a substantial effect on the area and euen central moments of Gaussian chromatographic peaks. It was also shown that the odd central moments (peak position, skew) of symmetric peaks are not altered by detector nonlinearity. These “chromatographiic” characteristics of the effect of a nonlinear detector on a peak contrast sharply with the effect of a nonlinear isotherm[, For example, an isotherm of the Langmuir type causes the peak centroid to shift to longer time and the peak to become increasingly tailed as the sample concentration is increased. A very general model of peak linearity based on the use of an infinite power series relationship between the sample concentration in the detector and the detector response was employed in the previous paper. The purpose of the present work is to demonstrate that such effects are indeed observable under reasonable chromatographic conditions. We chose to investigate the effect of absorbance detectors of the type used in liquid chromatography based on the report of Stewart (2) whose calculations indicate that such detectors can be rather nonlinear even at low absorbances if the slit widths are too wide relative to the width of the spectral band. Many other types of detectors, e.g., the fluorescence (3) and refractive index devices ( 4 ) are known to be nonlinear but are not as commonly used as is the absorbance detector. There are a number of potential sources of nonlinearity in photometric detectors. Assuming that chemical nonlinearity due to solution equilibria (dimerization, etc.) as well as nonlinear column effects (sample decomposition, irreversible adsorption, nonlinear isotherm, etc.) can be dismissed, then the chief somces of nonlinearity in the context of LC detectors will be stray light and lack of source monochromaticity (2). Of course, the electronics per se may be nonlinear, for example, amplifiers may saturate and logarithmic conversion circuits may not function perfectly. Indeed the effect of photoamplifier bias will act to cause nonlinearity in the same way as stray light. Although it is possible to arrive at a closed form expression for the effect of stray radiation on detector nonlinearity, the relationship in the case of polychromatic radi-
ation is not available in closed form. I n this instance the relationship between the observed absorbance and the sample concentration will depend upon the wavelength, the spectrometer slit function, as well as the shape, width and position of the absorption spectrum of the sample. Nonlinearity will be greatest when the slits are wide and when the solution absorbance is a strong function of wavelength, Le., when dA/dX is large. It is also evident that no matter how polychromatic the radiation is, the limiting value of the absorbance (at infinite sample concentration) will be set by the stray radiation and photoamplifier bias currents. Thus the effects of stray radiation and polychromatic radiation must be considered simultaneously. Due to the above complications we decided not to generate theoretical calibration curves such as those shown by Stewart (2). Rather an experimental calibration curve was generated and then the peak moments were computed by assuming that the concentration profile at the detector was Gaussian. EXPERIMENTAL SECTION Instrumentation. All experiments were carried out with a Perkin-Elmer (Norwalk, CT) Series 3B liquid chromatograph operated in the isocratic mode (80% acetonitrile and 20% water by volume) at a flow rate of 1.0 mL/min. A Perkin-Elmer LC-55 variable-wavelength detector was used for all results reported here. This instrument has a bandwidth of 2 nm and a cell path length of 6 mm. The stray light specification of the instrument is skated to be 0.1% at an absorbance of 220 nm when used with a conventional cuvette and not an HPLC flow cell. The manufacturer does not provide the stray light figure of the flow cell plus the detector. All measurements were carried out at 212 nm (see Figure 1). The column employed in this work was 12.5 cm long X 4.6 mm in diameter and was packed with 10 pm Lichrosorb RP-18. Samples were introduced to the column via a Rheodyne 7125 sample injection valve fitted with a 20-pL loop. The volume of sample was defined by the loop size. The signal was recorded simultaneously on a 10-mV recorder and transmitted to a data acquisition system comprised of a high-speed analog to digital converter and laboratory computer system (Data General Corp., NOVA 11). In all the experiments reported here care was taken to ensure that no component of the detection system or the data system was saturated; thus the nonlinear effects reported here are spectroscopic in origin and not electronic. It should be noted that the amplifier bias currents are totally negligible with this detector; Le., with the light source totally blocked the apparent absorbance is much greater than the maximum values observed with a sample in place. Reagents. The sample in all cases was biphenyl (Aldrich, Milwaukee, WI) and was used without further purification, Samples were dissolved in the mobile phase. Water was purified by use of a Continental demineralization system, it was then passed over a column of XAD-2 resin (Rohm & Haas, Philadelphia, PA) to remove trace organics and passed through a 0.5 pm Millipore filter prior to use. HPLC grade acetonitrile (Mallinckrodt, Paris, KY) was used in all experiments. Computations. All data were obtained at a rate of 150 ms per point. Each concentration was run at least three times. The various concentrations and individual runs at each concentration were run in random order. The moments reported here were
0003-2700/8110353-1373$01.25/00 1981 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
Table I. Effect of Sample Concentration on Measured Peak Characteristics
a
concn,O mM
peak height, A
peak area, A . s
centroid ( m 5 )s,
0.1 0.5 1.o 2.5 5.0 10.0 25.0
0.187 i 0.02 0.810 i 0.01 1.39 t 0.02 2.46 f 0.02 2.62 ? 0.04 2.66 0.005 2.68 i 0.03
1.580 i 0.02 6.912 i 0.1 12.08 i 0.1 24.98 i 0.6 33.43 i 0.6 40.38 i 0.4 49.34 i 0.9
202.9 f 0.2 202.8 i 0.2 202.7 f 0.2 202.4 i 0.4 202.9 ? 0.3 203.0 e 0.4 203.9 i 0.5
+ .
second central s2 moment (Z,), 11.8 i 12.0 i 12.7 f 14.2 i 19.1 -i 25.5 i 39.6 i
0.6 0.1 0.08 0.5 0.8 0.7 1.9
Concentration iniected. 30 I
CS
IC
005 AU
A
300
200
WAVELENGTH (nm)
Figure 1. UV spectrum of biphenyl. Conditions: 0.01 mM solution in 80:20 acetonitrile/water. Note the position of the 212-nm wavelength on the spectrum.
obtained by simple statistical summation techniques. Peak height was computed by fitting the topmost 15 data points to the equation of a parabola via Gram polynomials as described by Rogers (5). We also attempted to avoid the enormous problems caused by noise in the measurement of the higher central moments (6) by use of the Yau algorithm for asymmetric peaks (7). We found, as anticipated, that this approach failed because of the lack of fit of the peak shapes (see Figure 2) to the exponentially modified Gaussian model which is the basis of Yau's approach. Thus we are not able to report the third and fourth central moments due to their extreme imprecision.
RESULTS A spectrum of biphenyl is shown in Figure 1. All chromatographic results were obtained on the shoulder at 212 nm in order to maximize the effect of nonlinearity. The calibration curve (Figure 3) is evidently extremely nonlinear a t concentrations in excess of 1.0 mM. Note that the absorbance is only slightly in excess of 1.4 when a noticeable nonlinearity occurs in the calibration curve. A theoretical calibration curve can be computed on the basis of eq 1which describes the effect
of stray monochromatic radiation. In this equation, A is the true absorbance of the sample, 0 is the ratio of the intensity of the stray radiation to the intensity of the monochromatic radiation incident upon the sample cell, and AoM is that which would be measured by the electronics in the presence of the stray radiation. The fraction B is easily calculated from the limiting value of the observed absorbance at very high concentration. For our experimental system B calculates to be 2.1 x A theoretical calibration curve (not shown) based on eq 1 and the experimental estimate of the stray light indicate that stray light is not the sole source of nonlinearity, i.e., nonmonochromatic light is acting to decrease the observed absorbance below that expected based on the stray light effect.
B
C
D
E
L
Figure 2. Experimental chromatograms: curve A, 0.1 mM; curve B, 1.0 mM; curve C, 2.5 mM; curve D, 10 mM; curve E, 25 mM. All
conditions are described in the text. Note the changes in absorbance scales at 0.1 and 1.0 mM. A series of chromatographic peaks obtained at various concentrations of samples but otherwise under identical conditions are shown in Figure 2. Peak broadening is evident a t concentrations in excess of 2.5 mM. It is important to note that peak symmetry is not significantly altered until the sample concentration exceeds at least 10 mM. At a concentration of 25 mM a slight but definite tail is observed in the peak. This indicates that the partition isotherm is linear until rather high concentrations (see below). The experimental peak height, area, centroid, and second central moments are given in Table I along with estimates of their standard deviations. The response in terms of peak height is essentially constant from 2.5 mM upward but the peak area continues to increase because the peaks become progressively wider. This is in accord with the general results of the previous work (I) where it was shown that the relative change in the peak area was less sensitive to detector nonlinearity than is the change in peak height. The position of the peak centroid ( m J is remarkably reproducible in view of the fact that all samples were run in random order. As predicted by the previous work, the centroid of a perfectly symmetric peak is not altered by detector nonlinearity. The position of the centroid is essentially invariant from 0.1 to 10 mM sample concentration. I t could be argued that the calibration curve nonlinearity shown in Figure 3 is not due to the detector response but rather to a nonlinear chromatographic process, e.g., a nonlinear isotherm. This, however, is not the case since a direct and inescapable consequence of a nonlinear isotherm is first to distort peak symmetry and then broaden the peak and shift the position of the peak centroid (8). Inspection of the chromatograms of Figure 2 provides no evidence whatsoever of alteration of peak symmetry at sample concentrations less than 25 mM whereas the detector response is essentially flat (saturated) just above 2.5 mM. It might be argued that the nonlinearity of the calibration curve is related to irreversible adsorption of part of the sample. This position is also untenable because this effect would be
ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
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Table 11. Calculated Values of Peak Characteristics as a Function of Sample Concentration
1
3.0
concn, mM
0
A,=,= A
0.1 0.5 1.o 2.5 5.0 10.0 25.0
0.183 0.796 1.35 2.64 3.05 3.19 3.63
Computed from e q 5. Computed from eq 10. (I
area,b A.s
second central moment,c s2
1.40 7.01 13.8 25.3 34.6 40.7 47.7
11.9 11.9 12.6 14.8 17.9 22.6 29.4
Computed from eq 9.
1_
25.0
CONCENTRATION (mM)
Flgure 3. Experimental calibration curve for biphenyl: (curve A) expanded scale to show linearity at low concentration; (curve B) full data range. All conditions are described in the text.
0 c
'
/h
i
j
I
1 - i.F
m:
212 nm
0
202 nm
25-
B+ / ; A 0
,L---l-L
10
20
00
LoG,,Aca~c
Flgure 4. Results of the open tube experiment. All conditions are described in the text. Curve A (U) 212 nm (% deviation = (A,,,,d 3.90C)/3.90C). Curve B (0) 202 nrn ( % deviation = (A&d -
10.68C)/l0.68C).
most serious a t low concentration whereas our calibration curve is linear a t low concentrations and passes through the origin. Secondly, biphenyl is an ideal sample for reversedphase chromatography on an octadecylsilane bonded phase; it is totally aromatic and contains no heteroatoms which could interact strongly with bare silanol groups. Finally, the fact that the area is less effected by the calibration curve than is the height, as predicted by previous work ( I ) , argues against the loss of a fraction of the sample which is not eluted from the column. If a given amount of material does not elute and the remainder elutes via a linear isotherm, then peak area and height would be equally altered and would be proportional to one another. The fact that the effects reported here are not chromatographic in origin but are due to the detector was proved by replacing the column with a short length of HPLC tubing. The maximum absorbance of a series of dilute solution of biphenyl (0.05-10 mM) was recorded. These results are summarized in Figure 4. The percent deviation of the response from linearity was calculated as 1 0 0 ( A , ~- Aarlcd)/Acalcd and plotted vs. the decadic logarithm of the calculated absorbance. The calculated absorbance was estimated from the
known concentration and the molar absorptivity measured at a concentration of 0.05 mM. This figure shows a clear monotonic increasingly negative deviation from linearity. One should note that the detector is linear to higher absorbances a t 202 nm than at 212 nm which is an expected result due to the contribution of the nonmonochromaticity to detector nonlinearity when one works on a shoulder of a spectral peak. The above results indicate that the partition isotherm is quite linear up to 10 mM. The barely significant increase in the position of the centroid a t a sample concentration of 25 mM is in agreement with the development of a slight but definite tail in the peak (see Figure 2, curve E). It is clear that the pumping system is much more repro,ducible than is our ability to inject and quantitate the size of the signal. The precision of rnl at a given concentration, which as pointed out above were not run sequentially, indicates that the long term average flow rate is constant to approximately f0.2%. The relative standard deviation in peak height decreases with sample concentration. This is symptomatic of noise limitations. In contrast the relative standard deviation in area is approximately constant. The measurement precision is quite good enough to allow reproducible estimation of the second central moment. Table I indicates that the second central moment is constant where the detector is linear (0.1,0.5 mM) but increases significantly as soon as a negative deviation from linearity occurs (1mM). This is also in qualitative accord with our previous results (I). Although we calculated the third and fourth central moments from our data, the reproducibility was extremely poor and the slight trend in predicted by the general model of the effect of nonlinear detectors could not be verified. Table I1 shows the results of some predictions of peak height, area, and second central moment based on the assumption that the actual concentration profile in the detector is Gaussian a t all concentrations. If one assumes that at the lowest concentrations used the detector is linear (see Figure 3, curve A) and that the peak is Gaussian one arrives a t the following relationships:
6
A d t = cbCoVo/F
(3)
where c is the molar absorptivity, b the path cell length, C" the injected concentration, V" the volume injected, uv the square root of the variance due to the column per se (in volume units), and F the mobile phase flow rate. Chromatographic theory indicates for a pure Gaussian peak
m2 (time units) = (uV/n2
(4)
Thus for a Gaussian peak the above equations lead to
(5)
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ANALYTICAL CHEMISTRY, VOL.
53, NO. 9,AUGUST 1981
Values for A,, were computed from the measured peak area and second central moment a t each concentration and are summarized in Table 11. The first two entries (at 0.1 and 0.5 mM) are in good agreement with measured peak absorbance values and areas. Note that the second central moment a t these lowest concentrations is constant and corresponds to a a, of 57.5 pL. This agreement verifies that the peak is essentially Gaussian. The calculated value of A,, at 1.0 mM also agrees with the measured value but - only if one uses the measured second central moment, m2, which is definitely larger than the value a t the lower concentrations. Above 1.0 mM the agreement between calculated and measured values of the absorbance at the peak maximum worsens progressively as the signal peaks become less Gaussian (see Figure 2). On the basis of a value of 57.5 pL for a, and a 20-pL sample volume, one calculates from eq 6 that any sample will be
c,,
=
(6)
COVO/&r,
diluted to 0.138 times its initial concentration upon arrival at the detector. The calibration curve shown in Figure 3 can now be transformed to a plot of absorbance vs. the concentration of solute in the detector cell. This plot constitutes a graphical representation of the functional relationship between A and C as embodied in eq 7.
(7)
A = f(C)
The area and second central moment can be calculated assuming that the concentration as a function of time in the detector is Gaussian, i.e. C(t) = 0.138C0 exp[(t - m1)2/2a,2]
(8)
where ut2is the variance due to the column in time units. This leads to the results area =
A ( t ) dt =
Jrn
10
f ( C ( t ) ) dt
(9)
In the data reported in Table I1 the above integrals were carried out by use of a 16 point Gauss-Legendre integration technique (9). Values of the absorbance were estimated based on a piecewise linear interpolation between each data point. Values of ut were obtained from eq 11. The greatest dis-
ct =
a,/F
(11)
crepancy (14%) between a calculated and measured area occurs a t a concentration of 10 mM. The average absolute deviation is 5 % . The calculated second central moments are also shown in Table 11. Agreement with experiments is very good except for the highest concentration where the discrepancy is nearly 25%. We believe that this may be due to the tail which starts to develop at this very high (25 mM) concentration. We believe that the experimental work presented here indicates that chromatographic detectors can produce quite nonlinear calibration curves. Such nonlinearity leads primarily to distortion of the even central moments of symmetric peaks. Clearly detector nonlinearity is not an insurmountable analytical problem since one can rerun the sample after dilution to move to a linear part of the curve. Detector nonlinearity evidently has some impact on the experimental measurement of the resolution of overlapping peaks and in the area of steric exclusion chromatography can lead to erroneous estimates of molecular weight distributions. Obviously any attempt to build algorithms into chromatographic data systems to correct for nonlinear detectors must take into consideration the fact that nonlinearity will alter the peak's shape as well as its area and height. Finally, it should be noted that the detector used in this work has a spectral bandwidth of only 2 nm. Many other commercial variable wavelength LC detectors have bandwidths of 5 to 20 nm and therefore can potentially be nonlinear a t lower absorbances than reported in this work. LITERATURE CITED Carr, P. W. Anal. Chem. 1980, 52, 1746-1750. Stewart, J. E. J. Chromatogr. 1979, 774, 283-290. Hercules, D. M. Anal. Chem. 1968, 38(12), 29A-43A. Meggs, R. J. I n "Practical High Performance UquM Chromatography"; Simpson, C. F., Ed.; Heyden: London, 1976; p 279. Pauls, R. E.; Rogers, L. E. S e p . Sci. 1977, 72, 395-413. Chesler, S. N.; Cram, S. P. Anal. Chem. 1971, 43, 1922-1933. Yau, W. W. Anal. Chem. 1977, 49, 395-398. Littiewood, A. 6 . "Gas Chromatography, Prlnciples, Techniques and Applications"; 2nd ed.;Academic Press: New York, 1970; p 12. Atkinson, K. E. "An Introduction to Numerical Analysis"; Wiley: New York, 1978; p 231.
RECEIVEDfor review December 9, 1980. Accepted April 23, 1981. This work was supported in part by a grant from the National Institute of Occupational Safety and Health (5 R01 0B00876-02). Summer support for W.E.B. from the Uniroyal Corp. is gratefully acknowledged.
