Anal. Chem. 1990, 62, 1723-1730
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ARTICLES
Effects of Extracolumn Convolution on Preparative Chromatographic Peak Shapes Eric V. Dose and Georges Guiochon*
Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600,and Analytical Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
The effects of extracolumn components on preparatlve (nodnear) chromatographic peak shapes are Investlgated by convoktlng cokmn lnlet and outlet profflesofchromatogrsphic dmulatlons wlth Gausslan and exponentlal spreadlng functlons. Broadenlng of the Inlet proflle serves malnly to retard the elutlon of the peaks. The chromatogram base-llne wldth Is very rwMdtlve to maH amounts of broadenlng at the outlet (detector). WMle llnear chromatography Is modeled by slmple shlft-lnvarlant convolutlon, nonllnear chromatography re@res the more complex shm-vertant convoknlon model. This removes the property of varlance addltlvlty famlllar In llnear chromatography. The relatlve utllltles of three measures of preparatlve chromatographic peak wldth are tested by using convoluted peak shapes.
Peak shapes and widths are used differently in the analytical (linear) and preparative (nonlinear) modes of chromatography. In analytical chromatograms, the shapes of neighboring peaks are generally very similar, and their retention difference and their average width suffice to measure their resolution if their areas are at all similar. In preparative chromatography, there is no such comprehensive measure of peak separation because there is no universally applicable measure of peak width on which to base a measure of separation. Still, it is often important to compare preparative peak widths. As an extreme example, one may even obtain useful component enrichment when the two peaks differ only in width (I). The ways in which peak broadening due to the injection profile, the detector response, and the hydrodynamic mixing chambers (tubing, connectors, etc.) affect preparative chromatographicpeak shapes are complex but very important to the quality of such separations. We present here a study of the effects of extracolumn peak broadening on the widths of preparative (nonlinear)chromatographic peaks for the same reasons such studies were performed in analytical (linear) chromatography decades ago (2-4).The resulting variety of peak shapes obtained in the course of this study also allow us to comment on the relative validity of several peak width measures for preparative chromatography. It is generally agreed that chromatographic peak variances (and therefore calculated plate heights) contributed by the 0003-2700/90/0362-1723$02.50/0
injector, the column, the detector, and other components are additive in linear chromatography (2-4). When experimental variances are not additive, the discrepancy is traceable to incomplete radial mixing between components (5-7), to mobilephase decompression in gas chromatography (5),or simply to difficulties in computing the variance (8). Variance additivity is extremely useful in linear chromatography because, by knowing the column efficiency and the sum of the extracolumn variances, one can predict peak variances. This in turn allows one to identify the system component that most needs improvement. In nonlinear chromatography at low loadings, that is, when concentrations are low enough that the relevant portion of the sorption isotherm is practically linear, one may as a first approximation add the variance contributions due to kinetic (nonequilibrium) and thermodynamic (nonlinear) causes to give the total column variance contribution (9,lO).Though it seems reasonable to treat the effects of slight isotherm nonlinearity as a perturbation on linear chromatography (10-12),the accuracy of variance additivity under nonlinear conditions has never been tested satisfactorily, especially with respect to extracolumn effects. When the amount of solute loaded into a column becomes significant compared to the column capacity, column and extracolumn variance contributions to the variance of the resulting chromatographic peaks are generally nonadditive. Unfortunately, this information is not as useful as one might hope, for three reasons. First, variance additivity of chromatographic peak contributions has not been tested experimentally in the general case. Second, it is not clear that variance is the best measure of the width of preparative chromatographic peaks. Some other peak width measure may much better predict a given peak shape's expect value for preparative separation or enrichment. Third, it is unknown in generaljust how injection, column,and detedor convolution (dispersion) processes interact. This requires that we rely on intuition to decide, for example, whether a mixing chamber upstream from the column degrades a separation more or less then would the same chamber downstream. We don't believe that preparative chromatographers should labor under such absolute uncertainty about the effects of extracolumn dispersion on preparative chromatographic peak widths. Therefore, we present here a study of such effects. We are 0 1990 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 17, SEPTEMBER 1, 1990
preparing for later publication a similar study on two-solute chromatograms where solutesolute competition for stationary sites is likely to interact with extracolumn dispersion in very complex ways.
THEORY Convolution as a Model of Chromatography. The only completely general way to simulate the effect of chromatographic components on the resulting peak shapes is through convolution (13). Convolution and its inverse, deconvolution, have found use in chemical disciplines including crystallography (14,15),NMR spectroscopy (16),isotherm analysis (19, and electrochemistry (18)as well as outside chemistry in image processing (19),X-ray spectroscopy (13),and physically remote measurements (20). The convolution process most familiar to chromatographers is surely Savitzky-Golay smoothing (21) and corrections (22, 23). Like all classical convolution processes, it operates on an input curve (e.g., a chromatogram) to yield an output curve that is smoother and broader in precisely known ways. The terms convolution and deconvolution are used here in the original mathematical sense, not in the parochial sense often employed by chromatographers (24-29) when band-shape decomposition (30) or band-shape analysis (31) is actually meant. Classical convolution applies a spreading function S ( t)’ to a time-dependent input function F ( t ) yielding a resulting function R(t) according to the operation
R(t) = l m S ( t ?F ( t -t? d t -m
(1)
Since one cannot in general obtain analytical expressions for the input chromatograms F ( t ) , one must work with discrete data and vector operations. Thus, a chromatogram represented as a vector of concentrations F sampled at the uniform time interval Bt may be convoluted with a spreading function S, also sampled at the interval 6t, to give a resultant chromatogram vector R as m
Ri =
,
SitFi-2
(2)
tk-m
where indices outside the assigned range for S or F map to values to zero. This procedure, discrete convolution, is defined only for all S p and Fi finite and when the conditions in eqs 3-6 are satisfied:
5 S? 5 F~
i’=-m
finite
(3)
finite
(4)
I=--
mutative, and variance-additive, respectively (13). In linear liquid chromatography, all column segments and all time periods are equal in their effect on the distributions or the profiles of the compounds migrating through the column. This means that each theoretical plate’s effect on the concentration profile passing through it is indistinguishable from the effect of any other plate. For each plate, one can write a spreading function S (eqs 1 and 2) whose variance equals the variance added to the moving profile by that plate and whose first moment is proportional to the retention effected by that plate. Thus, linear chromatography is a n iterative shift-invariant convolution of profile shape. This metaphor has arisen before (32). The properties of shift-invariant convolution determine those of linear chromatography. They share variance additivity. Commutativity implies that, for example, a given physical mixing chamber should have exactly the same effect on chromatogram peak shapes whether it is placed upstream of the column, downstream of the column, or even between the two pieces of the bisected column. The exact peak shape that results is nearly Gaussian, which is known to be the limiting resultant function R as any spreading function S (for example, that arising from one theoretical plate) is applied repeatedly to an input function F (13). This limiting function if F*S*S*S ... = F ( * S ) N ,and the resulting variance is q2+ Nus2, where N is the number of times S is applied, i.e., the number of theoretical plates, and where uF2 and 6a2 are the variances of the functions F and S, treated as distributions along the time axis. Not only the variance but the entire peak shape may be computed from eqs 1and 2. For example, the commonly adopted EMG (exponentially modified Gaussian) peak shape is simply a convolution of an exponential function, due to an ideal mixing chamber such as a detector, and a Gaussian function, the linear chromatographic peak shape. The detailed effects of injector (3, 33) and detector (34) spreading functions in linear chromatography have been presented earlier. In nonlinear chromatography, the effect of each plate on the analyte profile depends on the concentration of the compounds in that plate. Except in the trivial case where the concentration in the column is uniform, the appropriate spreading function V is time-dependent. Thus, to describe nonlinear chromatographyas a convolution process, one must write
R ( t ) = l m V ( t ’ , t F) ( t - t? d t -m
or, for the discrete case, m
Ri = i‘=-- V(t)iiFi-i,
lim Si,= 0
(10)
(11)
1il-m
lim Fi = 0
lip-
The classical integral and discrete convolution operations of eqs 1and 2 are called shift-invariant, meaning that the same spreading function is applied to every time t or element i; that is, the spreading function is invariant with the shift applied to it. We abbreviate eqs 1and 2 as R = S*F. In this notation, three important properties of shift-invariant convolution are written as
R d t = XIS d t 1 - F d t -m
S*F = F*S
(7) (8)
that is, shift-invariant convolution is area-conserving, com-
where V ( t )is a time-dependent spreading function. Equations 10 and 11are examples of shift-variant convolution and may be abbreviated R = V(t)*,F, where we emphasize the nonclassical, shift-variant nature of the convolution by adding a t subscript to the operator symbol. The properties of shift-variant convolution are quite unlike those of the classical shift-invariant analogue. What is important for the present work is that shift-variant convolution is neither commutative nor variance-additive. Noncommutativity results simply because eqs 10 and 11 are asymmetric with respect to V and F. In a column, this means that the exact spreading functions applied to the inlet profile depend on the profile’s own shape. The simplest explanation for the variance nonadditivity is that the spreading function V(t,t’) has no uniquely defied variance-one cannot determine what variance contribution it will make to the resulting profile until one knows the actual inlet profile. In general, the limit of the
ANALYTICAL CHEMISTRY, VOL. 62, NO. 17, SEPTEMBER 1, 1990
iterative shift-variant convolution F*,V*,V*,V... = F(*,V)N is not Gaussian. These properties imply that the profile eluted from the column may depend on the column inlet profile in unexpected ways. One can write the operation performed physically by a nonlinear chromatographic system as R = D*[C(t)*,I] where C(t) is a shift-variant convolution function characteristic of the columnsolute interaction. It is important to note that while shift-invariant convolution properties allow one to add the variances of the detector spreading function D and of the column outlet profile C(t)*,I, shift-variant convolution properties prevent one from adding the variance of the column spreading function C(t) (whose variance is not uniquely defined) to that of the inlet profile I. One consequence of this nonadditivity is that a perturbation to the inlet profile that results in a variance increase Au? will not in general add Aut to the variance of the measured chromatogram R. Similarly, the peak shape and its variance will change if a mixing chamber downstream of the column is moved upstream, or vice versa; that is, in general C(t)*,[M*I] # M*[C(t)*,I]where M is the mixing chamber spreading function. Since only the column convolutes the profile in a shift-variant manner, components downstream of the column always exhibit classical, shift-invariant convolution properties including variance additivity. The same reasoning that applies to chromatographic components applies to segments of a column operated in the nonlinear mode. One could demonstrate by induction that a given column, viewed as a sequence of very short columns (or plates), will not generally produce peak variances equal to the sum of the kinetic and thermodynamic variance contributions (1&12,35,36). Of course, such additivity may hold approximately for systems close to isotherm linearlity (10-12, 35). In summary, the ways that the nonlinear and linear modes of chromatography produce peak shapes differ in the same ways that shift-variant and shift-invariant convolution properties differ. The results presented here employ this close correspondence to illustrate how concentration overload and extracolumn convolution interact to give highly nonlinear overall behavior. Measures of Preparative Chromatographic Peak Width. If all chromatographic peaks were uniformly shaped, all peak width measures that are scaled to the peak height would be equally useful. For Gaussian peaks that result from linear chromatography, one may choose a t one's convenience between the variance or the standard deviation (3,4,6-8,10, 37-41),the width a t 13.53% of the peak height (4u) (38),the minimum width enclosing 95.46% of the peak area (also 40) (%), the width at half-height (5,11,38,42), the width at 10% of the peak height (11),or the base-line width bracketed by tangents drawn through the peak's two inflection points (43). While the above measures are functionally equivalent when applied to noiseless Gaussian peaks, the same is not true when the widths of preparative chromatographic peaks must be measured. The variance, or second central moment, of the peak considered as a distribution on the time axis is a natural measure. However, nonlinear isotherms and the finite efficiency of real columns tend to produce tails on the peak shapes, and these tails may not contribute to the computed variance to the same extent that they contribute to the overlap with another solute peak. The variance can also be difficult to measure accurately because it is so sensitive to base-line placement (44). For these reasons, we have examined two additional peak width measures. The width a t 10% of maximum height, WIoH, has been used previously in the computation of plate heights of skewed Gaussian peaks (45, 46). This measure is easy to obtain for practically any peak
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and is relatively insensitive to base-line noise and drift. The minimum width including 99% of the peaks area, W,,, has obvious advantages when applied to preparative chromatography. The solute concentration in this width is the maximum possible with 99% yield. Also, for a peak shape with very extended tails of low concentration, as sometimes observed in preparative chromatography, WwAis quite sensitive to the shapes of these tails, just as the quality of a preparative separation would likely be, although this sensitivity makes W,,, more susceptible to base-line drift and noise. For the triangular approximation to peak shapes resulting from chromatographic systems exhibiting a Langmuir isotherm (10, 47), W1, and W,, are equal.
EXPERIMENTAL SECTION Semiideal chromatographic simulations were performed by Godunov integration as previously described (47-49).Each simulation required an injection profile, an isotherm, a solvent flow rate, and a column plate number and yielded an elution profile, that is, a vector of eluted component concentrations at uniform time increments after injection. The elution profile sample points were interpolated by an Akima spline function (IMSL, Houston, TX), and the values were stored at intervals of 1%of the void time t,, which is the unit in which all times in this work are expressed. Discrete spreading-function vectors were constructed from points, equally spaced in time, representing either a Gaussian or an exponentialspreading function of desired variance. This vector was extended sufficiently to include all values greater than lo4 of the peak value and was then normalized to a sum of 1. Discrete convolutionswere performed either on the injection profile, which always began as a pseudo-Dirac pulse of only one plate width (correspondinggenerally to a few milliseconds), or on the simulated elution profile. The convolution result was accepted only if the input and output profile areas, computed as the area under the Akima splines drawn through all (time, concentration) points defined by each vector, agreed in sum to 0.02%. Agreement was generally to 0.002%. Zero time for both the Gaussian and exponential spreading functions refers to the function maximum. The variances were computed by multiplying the concentration of each chromatogram point by the square of the difference between the point's time and the peak's first moment, constructing an Akima spline through the resulting points, integrating, and dividing this abstract area by the area of the chromatographic peak. WloH,the full width at 10% of the maximum height, was computed by constructing an Akima spline through the (time, concentration) points representing the chromatogram and determing by Newton approximationthe f i t and last times at which the spline value equals 10% of the maximum concentration. To compute WwA, the minimum width including 99% of a peak's total area, the Akima spline passing through all the (time, concentration) points representing the chromatogramand its total area was computed. Since the time boundaries of the minimum width necessarily have identical concentrations,the concentration defining the minimum width was found by iterative Newton approximation rather than by determining both boundary times directly. This iterative solution on concentration was much more stable numerically than was the solution on time. The initial concentration estimate was 10% of the peak maximum concentration (the exact solution for triangular peaks). All computation was performed to double precision (16 decimal digits) by using VAX FORTRAN on Model 8800 computers (Digital Equipment Corp.). Numerical subroutines from the Numerical Algorithms Group, Limited, and from IMSL were used where possible. RESULTS AND DISCUSSION Chromatographic peak shapes obtained under nonlinear conditions are largely dominated by the shape of the sorption isotherm describing solute distribution between the stationary and mobile phases. This dependence of peak shape on isotherm shape becomes more exclusive if the isotherm becomes more nonlinear, if the column efficiency is increased, or if extracolumn broadening is decreased. When the isotherm is
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ANALYTICAL CHEMISTRY, VOL. 62,
NO. 17, SEPTEMBER 1, 1990 8
i 0.06
4
I\
Lf=a'a'
n i
0.02 -
L'=O
.z.-
0.016 -.
5
E
6
3
1
5 Time I t ,
e-(t
')2/2002
(13)
In the other limiting case, a single, large mixing chamber that is perfectly mixed internally introduces a shift-invariant convolution whose normalized spreading function is an exponential decay given by for s(t?=
,,r,
t'< 0
for t f 2 0
-
3
1
concave downwards and has an upper limit of solute sorbed per m w of stationary phase (its capacity), as for the Langmuir isotherm model, the fraction of this capacity actually injected is the loading factor Lp As seen in Figure 1,given an efficient (No= 5000 plates) column, a narrow injection pulse, and no extracolumn broadening, the resulting peak shape elutes earlier on the average and broadens as Lfincreases. At higher Ln the earlier elution is due to the lower isotherm slope at higher concentrations. This lower isotherm slope results in a lower capacity factor k' and thus less retention. The broadening a t higher Lf is, in simple terms, due to the nonuniformity of the isotherm slopes experienced by the zones of different concentrations. In real chromatographs, there are additional, extracolumn contributions to peak broadening. We may divide these contributions into two limiting cases. In the first, a series of very many infinitesimally small chambers of any description will, because the product of any shift-invariant series tends toward a Gaussian function introduce a shift-invariant convolution whose normalized spreading function is a Gaussian function of variance uG2 as (2")'/2,G
0.008
7
Figure 1. Chromatographic peak shapes resulting from Langmuir isotherms and varbus loading factors L,. Conditions: k', = 5, nominal efficiency No (at k',,) = 5000 plates.
S(t? =
0.012
(14)
where T is the characteristic mixing time, equal to the chamber volume divided by the flow rate. In the following discussions, we present the effects of these two types of extracolumn convolution on the widths of preparative chromatographic peaks. Gaussian Extracolumn Convolution. When placed upstream of a column operated in the preparative mode (Langmuirisotherm,Lf = 0.20),a chromatographic component introducing Gaussian convolution serves mainly to retard elution without major peak shape changes, as illustrated in Figure 2. This behavior is completely different from the consequences of the same extracolumn convolution on linear (Gaussian) chromatograms where the retention time is unchanged but the peak broadens about its center. The peak width in the nonlinear case changes little because broadening due to isotherm nonlinearity overwhelms that due to the inlet profile shape. The retention time increases because the ob-
5 Time It,
7
Flgure 2. Effect of Gaussian injector convolution on nonlinear peak shapes. Conditions: k', = 5 N o = 5000 plates. Gaussian u values as a fraction of t, and in order of decreasing peak height: 0, 0.120, 0.224, and 0.370.
n
Time It,
Figure 3. Effect of Gaussian detector convolution on nonlinear peak shapes. Conditions and Gaussian u values as in Figure 2.
served retention time near zero concentration, that is, near the end of the observed peak tail, is related to tI,O,the time at which the inlet concentration falls to near zero as t1,o
= tM(1
+ 129 + t1,o
(15)
where tM is the column void time. With increasing Gaussian convolution variance, tI,Oincreases even though the center time of the Gaussian peak is fixed. This retardation has been noted before for broad rectangular injection profiles a t very high column efficiency (50). These effects are minimized in most preparative chromatographsby pumping the sample solutions directly into the separation system, eliminating the need for injection loops and the resulting zone broadening (6, 7). In summary, upstream Gaussian convolution causes slight broadening and significant retardation of preparative chromatographic peaks. Downstream of the column, Gaussian convolution broadens the detected chromatographic peak as illustrated in Figure 3. The sharp front is smeared to the extent that one could not determine without deconvolution that a front had existed in the peak as eluted from the column. The descending part of the peak and the retention time at which the concentration approaches zero are largely unchanged by downstream Gaussian convolution for a loading fraction of 0.20. As expected from the properties of shift-invariant convolution,the first moment is unchanged and the peak variance is increased by exactly the downstream spreading-function variance, just as found in linear chromatography. The variance of a nonlinear peak is affected in complex ways by Gaussian convolution (Figure 4). A t zero Lf, the special case of linear chromatography,the variances are additive. T h e
ANALYTICAL CHEMISTRY, VOL. 62, NO. 17, SEPTEMBER 1, 1990
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2
speak
W OH D 6 -
- Lf=o.lo
I
1::,
0
4 -
I
2
---
w
h
h
o
Lf'O
8
Flgure 4. Peak variance as a function of extent of Gaussian convolution. I: injector. D: detector. Conditions as in Figure 2.
Figure 5. as a function of extent of Gaussian convolution. I: injector. D detector. Conditions as in Figure 2.
curves representing peak variance uM2 vs Gaussian variance
presence of tails that are significant in concentration. As displayed in Figure 5, injector and detector convolution Wlw values are identical for Lf of zero. The slope of WloH2vs uG2 expected for Gaussian peaks, 8 In 10 N 18.421, is obtained. At positive Lf, neither curve is linear since the peak shapes are not Gaussian, and the variance and Wlo,2 need not be proportional. As expected from Figures 2 and 3, when Lf is positive, WloH increases rapidly with detector UG but more slowly with injector uG. Thus,Wlw appears to be more useful than the variance as a measure of the width of preparative chromatographicpeaks convoluted with a Gaussian spreading function. Another peak width measure, WSA, is the minimum width including 99% of the peak area. This measure is significantly more difficult to compute than are the variance and Wlw, but it may be more useful because WwA is much less easily fooled by long tails of very low concentration. For linear chromatography (Lf = 01, the WSA values on the injector and detector sides are identical, and the slope of WSA2 vs uG2 is that expected for Gaussian peak shapes; that is, 4s2 N 26.539 where erf (s/2'I2) = 0.99 (Figure 6). Otherwise, the dependence of WwAon Lf and UG is qualitatively similar to that of Wlom Exponential Extracolumn Convolution. Upstream of the column, the effect of exponential convolution on preparative peak shapes is very similar to that of Gaussian convolution-slight broadening and significant retardation (Figure 7). For equal spreading-function variances, exponential convolution is somewhat more effective than is Gaussian convolution (as in Figure 2), particularly in the duration of the low-concentration tail caused by long characteristic times T . Downstream of the column, exponential convolution both retards and broadens the peak (Figure 8). Retardation, expressed as an increase in the peak's first moment, results from the nonzero first moment of the exponential spreading function. (Since the Gaussian spreading function has zero first moment, it causes no retardation when downstream of the
uo2 are superimposed for convolution upstream and down-
stream of the column and are linear with a slope of exactly one. Under all preparative conditions (Lf of 0.01-0.20) examined, upeak2is still linear with .a2 of downstream (detector) convolution, as expected for shift-invariant convolution. The effect of upstream (injector) convolution is more complex. At low but nonzero Lf, the injector variance contributes less to uw2 than does the detector variance. However, at high Lf, the opposite is true. At very small injector variances at high Lf, u w 2 is extraordinarily sensitive to uG2. At high Lf, the great decrease in injected concentrations caused by even a small injector variance causes a broad distribution of the k' values and thus of the velocities within the zone. This distribution of the velocities results in broadening of the observed peak. At intermediate loadings (Lf of about 0.05), the injector-side and detedor-side curves actually cross. Comparing Figures 2 and 3, the peak shape resulting from downstream Gaussian convolution is more detrimental than that resulting from Gaussian convolution upstream of the column since the former clearly risks significant interference with other eluting peaks more than does the latter. However, this result is absolutely contrary to the conclusions one draws by comparing the u W 2curves of Figure 4. The inescapable conclusion is that for purposes of evaluating real separations, the peak variance is not a universally valid or even useful measure of the width of a preparative chromatographic peak. Though, at very low loading factors, the variance may still have limited utility, one cannot know accurately for what combination of the loading factor and the extent of convolution (if any) the variance is the best measure. We have in the course of this work examined other peak width measures that may more closely measure the quality of preparative peak shapes for real separations. The peak width measured a t 10% of the maximum peak height, W l o ~is, attractive because it is easily measured numerically or graphically and because is accounts for the
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 17, SEPTEMBER 1, 1990 18
,
0.024
1
p,
1
3
5
7
Time / ,t
Figure 8. Effect of exponential detector convolutionon nonlinear peak shapes. Conditions and characteristic T values as in Figure 7. 1.2
o >L f =o , ,
-0.04
I
,’:,
1
, , , , , , , ,
/, 0
0.04
0.08
c
0.12
Flgwe 8. WgpAas a function of extent of Gaussian convolution. I: injector. D: detector. Conditions as in Figure 2.
6
0.024
4
0.02
4
8
2peak 0.6
04
Lf = a 2 0
L f =0.10
0016 -
2 +
E
0.8
1
0012
1
Lf=0.05
0.2
0.004 { 0008
9=0.02
I
3
5 Time It,
Lf=O
0
14
0
0.04
0.08
0.12
0.16
7
Flgure 7. Effect of exponential injector convolution on nonlinear peak shapes. Conditions: k10 = 5, N o = 5000 plates. Characteristic T values as a fraction of t , and in order of decreasing peak height: 0, 0.120, 0.224, and 0.370.
column.) Broadening caused by exponential convolution downstream of the column appears less severe than that introduced upstream. Though the shape of the peak is affected more by downstream convolution, the downstream result appears preferable for purposes of preparative chromatography. The variance as a measure of peak width (Figure 9) agrees with this ordering of utility. As they must, injector and detector convolutions result in the same variance increases under linear conditions (Lf = 0). Under preparative conditions (Lf > 0), injector convolution introduces much more eluted peak variance than does detector convolution. This is the opposite of the findings for Gaussian convolution (Figure 4),but it is in agreement with the apparent widths of the actual peaks (Figures 7 and 8). The effects of exponential convolution of WloH are almost equal upstream and downstream of the column under all
‘2xp Figure 9. Peak variance as a function of extent of exponential convolution. I: injector. D: detector. Conditions as in Figure 7.
preparative conditions investigated (Figure 10). The fact that upstream exponential convolution appears more detrimental to preparative separations (Figures 7 and 8) as well as the ubiquitous nature of mixing chambers weaken the case for adopting WloHas a general peak width measure for preparative chromatography. By contrast, W,, is increased much more by upstream exponential convolution than it is by the same convolution downstream (Figure ll),in agreement with apparent widths. The sensitivity of W,, to the presence of long, low-concentration peak tails (as in Figure 7) is confirmed in the steepness of the WggA2 vs T~ curves a t small but nonzero upstream convolution 7. Choosing a Peak Width Measure for Preparative Chromatography. The most useful measure of chromatographic peak widths will consistently rank peak widths in the same order as the extent to which the widths are likely to cause overlap with other peaks. For the measure to be useful, it must include the effects of extracolumn broadening. We have not
ANALYTICAL CHEMISTRY, VOL. 62, NO. 17, SEPTEMBER 1, 1990
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12
10 14 -
D I
12
8
W:OH
10
D
6
I
/ D
S D I
4
D 4
2 2
0 4
0
0.04
0.08
0.12
(
i
2
Texp
0 -0.04
0
0.04
0.06
0.12
C
6
2
7exp
I: injector. D: detector. Conditions as in Figure 7.
Figure 11. WssAas a function of extent of exponential convolution. I: injector. D: detector. Conditions as in Figure 7.
included the effects of solute-solute competition for the retentive sites in the chromatographic column-we will do so in later studies. We have not identified an ideal width measure for nonGaussian peaks. Indeed, a measure that is ideal in the sense that the variance is an ideal measure of the widths of a Gaussian peak cannot exist since the severe shape constraints on Gaussian peaks allow the variance to describe the peaks shape more exactly than any measure could describe the shape of peaks of relatively unconstrained shape. Therefore, we turned our search from ideal peak width measures to useful measures. Measures that clearly fail to represent the utility of a peak shape for preparative chromatography for any reasonable loading factor Lf, say, 0-0.20, for either limiting case of convolution shape, Gaussian or exponential,or for any reasonable extent of convolution, say, of OG or T of 0-0.3tm, cannot be considered general. Choosing a general measure thus begins by eliminating unsuitable candidates. The full base-line width of a chromatographic peak is an unsuitable candidate for the simple reason that it does not exist for peaks derived from columns of finite efficiency in which cases the peak tails approach the base line asymptotically. To avoid this difficulty for real chromatograms, one could define the peak width as the time span during which the signal is greater than the chromatogram noise. However, such measured peak widths depend strongly on the detector noise level, which has nothing to do with the degree of chromatographic separation, and so this measure must be rejected, too. Drawing tangents through the peak shape’s two inflection points, a method very popular in linear chromatography,could be extended to nonlinear chromatography. There are, however, two difficulties. First, for very efficient columns with little detector peak broadening (see Figures 1, 2, and 7), a desirable case, the peak has only one identifiable inflection point. One could substitute the tangent just after the peak
maximum for the missing inflection point. However, doing so will cause the second problem-tails that extend far beyond the peak maximum are utterly unseen by this measure even though they seriously degrade preparative separations. Thus, tangent methods appear unpromising. The peak variance, the measure most closely associated with the theory of linear separations, does not generally reflect the likely degree of overlap with other peaks that is paramount in preparative separations. In Figure 4 (Lf= 0.20), the variance suggests that injector convolution is more detrimental than detector convolution when the peak shapes themselves clearly indicate the opposite (Figures 2 and 3). The ineffectiveness of the variance measures also means that the apparent plate heights and plate numbers calculated from the peak variances are not the best measure of column efficiency for preparative chromatography, except perhaps at low loading factor Lf, frequently the least economically attractive cases. The peak width measured at 10% of the maximum peak height, WloH,is insensitive to the presence of long peak tails of less than 10% of the maximum concentration. However, WloHis very easy to compute and works well for most peak shapes one is likely to encounter. Further, the peak tails in preparative chromatography tend to be compressed by later-eluting compounds (and by all compounds in displacement chromatography). If there is not such compression, then separation is already adequate, and the measures of the peak width may not be very important. Thus, without strong exponential convolution downstream of the column, which most chromatographers would take pains to avoid anyway, WIOHshould perform adequately without computational difficulty. The only peak width measure that always performed well in this study is W, the minimum width including 99% of the peak area. It is, however, rather difficult to compute accurately, particularly if the peak shape approaches the base line asymptotically. This difficulty is unavoidable-the susceptibility to base-line drift and other noise is exactly what
Figure 10. W I o Has a function of extent of exponential convolution.
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 17, SEPTEMBER 1, 1990
makes small amounts of overlap between non-Gawian peaks difficult to measure in the first place. All measures that attempt to account for long peak tailswill suffer this difficulty. Of such measures, W,A describes what the preparative chromatographer wants to know; that is, over how much time must a single eluting compound be collected in order to achieve high (empa, '"1 yield? In Other words, W%A the risk that a given compound is interfered with by other eluting compounds. Of the measures examined here, W,, appears to be most appropriate peak width measure for preparative chromatographic peaks, although W,, is nearly as good in most cases and is much easier to compute.
CONCLUSIONS Classical, shift-invariant convolution is a model for linear chromatography*The former process and the latter physical process share important properties including variance additivity. The more complex shift-variant convolution, applied iteratively at each theoretical plate, is a model for nonlinear chromatography. Variance additivity is not a property of either. Mixing chambers upstream of a column operated in prepmode retard peak A series Of mixing chambers that approximate a Gaussian spreading function of variance 2 is not expected to degrade a preparative separation as much as does a sing1e mixing chamber spreading function) of characteristic time 7 = 6. Mixing chambers downstream of a column convolute the column profile in a shift-invariant manner. Thus, detector convolution is the same process in linear and nonlinear and the variances introduced the column and the detector are strictly additive in both modes. The p e d variances and t h w the plate heighb and the numbers computed from the variances are not the best measures of preparative chromatographic separation efficiency. W,, is much better but is misleading at high column loading and when detector dispersion is strong and exponential in form, a situation normally avoided by chromatographers. W,, is the most reliable measure of preparative chromatographic peak width studied in this work, but we found it the most difficult to compute. ACKNOWLEDGMENT We the University Of Tennessee Computing Center's support of the computational effort. LITERATURE CITED (1) Dose, E. V.; Guiochon, G. Anel. Chem. 1990, 62, 174-181. (2) GuioChMi, G. J . GRS chrometogr. 1984, 2 , 139-145. (3) Sternberg. J. C.A&. C h f m t o g r . ( N . Y . ) 1966, 2 , 205-270. (4) Huber, J. F. K.; Riui, A. J . Chromstogr. lS87, 384. 337-348. (5) Maynard, V.; Grushka, E. Anal. chem. 1972, 4 4 , 1427-1434. (6) Golay, M. J. E.; Atwood, J. G. J. Chromatogr. 1979, 786, 353-370. (7) ~twood,J. G.; @lay, M. J. E. J . chr0metogr. 1981, 218, 97-122. (8) Lenhoff. L. J J. Chromtogr. 1967,384, 285-299.
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RECEIVED for review March 6,1990. Accepted May 3,1990. This work has been supported in part by Grant DE-FGOB86ER13487 from the U.S. Department of Energy, Office of Energy Research; by Grant CHE-8901382 from the U.S. National Science Foundation; and by the cooperative agreemerit between the University of Tennessee and Oak Ridge National Laboratory.
