Anal. Chem. 1991. 63. 120-130
120
raffin-naphthene separation by the molecular sieve column appears limited to carbon number 13 as the upper limit. Our attempts to recover paraffin mixtures containing n-paraffins greater than carbon number 13 consistently gave unacceptable low recoveries. Analytical precision of the system was evaluated using a practical naphtha and found to be acceptable. Typical results, given in Table I11 for six replicate measurements, show relative standard deviation (RSD) values for individual hydrocarbon types of 2.8% or less, with the majority of the various hydrocarbon types observed to be less than 1% RSD.
(2) ASTM D2789-81, Standard test method for hydrocarbon types in lowolefinic gasoline by mass spectrometry. ASTM Standards: ASTM: Philadelphia, PA, 1984; Vol. 05.02. (3) Suatoni, J. C. I n Chromatography in Petroleum Analysis, 1st ed.; Altgelt, K. H., Gouw, T. H., Eds.; Marcel Dekker, Inc.: New York, 1979; pp 121-136. (4) Miller, R. L.; Ettre, L. S.; Johansen, N. G. J. Chromatogr. 1983, 259, 393-412. (5) Leveque. R. E. Anal. Chem. 1967, 39, 1811-1818. (6) Merchant, P. Anal. Chem. 1968, 4 0 , 2153-2158. (7) Stuckey. C. L. J . Chromatogr. Sci. 1969, 7 ,177-181. (8) Whittemore, 1, C. Chromatography in Petroleum Analysis, 1st ed.; Marcel Dekker, Inc.: New York, 1979; pp 41-74. (9) Johansen, N. G.; Ettre, L. S.; Miller, R. L. J. Chromatogr. 1983, 256, 393-41 7. (10) Boer, H.; van Arkel. P. Chromatographia 1971, 4 ,300-308. (11) Stockinger, J. H.; Callen, R. B.; Kaufman, W. E. J . Chromatogr. Sci. 1978, 16, 418-426. (12) Boer, H.; van Arkei, P.; Boersma. W. J. Chromatographia 1980, 73, 500-512. 13) Huber, L. Chromatographia 1982, 16, 282-285. 14) Boeren, E.; van Henegouwen, R. B.; Bos, I.; Gerner, T. H. J . Chromatogr. 1985, 349,377-384. 15) van Arkel, P.;Beens, J.; Spaans. H.; Grutterink, D.; Verbeek, R. J . Chromatog. Sci. 1987, 25, 141-148. 16) Di Sanzo, F. P.; Lane, J. L.; Yoder, R . E. J . Chromatogr. Sci. 1988, 26. 206-209. 17) Di Sanzo, F. P.; Giarrocco, V. J. J . Chromatogr. Sci. 1988, 26, 258-266. 18) Curvers, J.; van der Sluys, P. J . Chromatogr. Sci. 1988, 26, 267-270. (19) Szakasits, J. J.; Robinson, R. E. U.S. Patent 4,534,207, Aug 13, 1985. (20) Brunnock, J. V.; Luke, L. A. Anal. Chem. 1968, 4 0 , 2158-2167. (21) Halasz, I.; Horvath, C. Nature 1966, 797,71-72.
CONCLUSIONS The multidimensional RPA has been shown to be a sensitive and reliable instrument for characterizing the composition of different types of catalytic reformer feedstocks and products in terms of paraffins, naphthenes, and aromatics by carbon number in the range C3-Cl2, with differentiation between C5-ring and C6-ring naphthenes through C8. The use of hydrogen as the carrier has produced an instrument with a high degree of accuracy and precision. Notable improvements are found in the separation of paraffins and naphthenes by carbon number by the use of a specially prepared molecular sieve 13X wall-coated capillary column.
LITERATURE CITED (1) Gary, J. H.; Handwerk, G. E. Petroleum Refining, TechnologyandEconomics. 2nd ed.; Marcel Dekker, Inc.: New York. 1984; Chapter 6.
RECEIVED for review .June 7, 1990. Accepted October 16, 1990.
Peak Shape Analysis in Sedimentation Field-Flow Fractionation Pierluigi Reschiglian, Gabriella Blo, and Francesco Dondi* Department of Chemistry, University of Ferrara, V. L . Borsari, 46, I-44100 Ferrara, Italy
Peak shape effects in sedimentation field-flow fractionation (SFFF) are investigated by Edgeworth-Crambr (EC) ieastsquares peak profile fitting, numerical integration, and graphical analysis. The use of an EC series expansion procedure with regard to the description of an SFFF process is discussed. The onset of nonlinear effects and secondaryorder phenomena such as steric contributions in the normal SFFF elution mode and injected sample overloading are detected by means of the EC series expansions through fitting patterns and Statistical peak shape parameter analysis. It has been found that, under Ideal conditions of monodisperse sample analysis, current SFFF instrumentation produce symmetric peaks with skewness values as low as 0.2, leading to accurate particle parameter estimations. Therefore, peak shape markers are a complementary check for detecting mixed elution processes, overloading effects, or polydispersity contributions that could affect the accuracy of particle parameter estimations. Peak shape analysis is generally able to allow for the unbiased determination of peak parameters at the onset of a steric contribution in the normal elution mode. Fast, handy rules are given for checking “pure” normal elution modes and the extent of bias on retention parameters.
INTRODUCTION Sedimentation field-flow fractionation (SFFF) is a wellknown, highly developed subtechnique belonging to the broad, 0003-2700/91/0363-0120$02.50/0
rapidly expanding family of field-flow fractionation (FFF) methods. Over 11 years ago, it was demonstrated that SFFF is capable of separating a wide variety of colloidal particles a t high levels of resolution (1-6). Fractionation in an SFFF experiment results from a sedimentation equilibrium superimposed on a steep mobile-phase velocity gradient. Such a combination of sedimentation field and longitudinal, laminar flow makes it possible to classify SFFF, like classical chromatography, as a “perpendicular field/parallel flow” separation technique (7). The output signal of both SFFF and classical chromatographic (8) experiments can be analyzed by analogous numerical methods. As for chromatography ( S l Z ) , peak shape analysis in SFFF is to be considered a basic tool for both fractionation channel performance characterization and physicochemical quantity measurement from band displacement and broadening. In its most improved version, peak shape analysis might not only yield meaningful shape parameters but also the best achievable approximation of peak profile shape, although strong shape asymmetry and tailing may be present. In FFF studies, to date, only the first two statistical moments of the eluted band have been referred to. Maximum peak position and graphical computations of the peak standard deviation by half-height peak width measurements have currently been employed for efficiency measurements (13,14). In the most currently applied theoretical and experimental approaches to FFF, the Gaussian peak approximation shape has been widely validated by the experimental results given in some fundamental papers regarding SFFF polymeric latex characterizations (1Fj-20). 1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
Unlike classical chromatographic applications, a more advanced numerical approach to statistical peak measurements and peak profile fittings has not, as yet, been required in FFF. However, as new FFF subtechniques have sprung up, secondary-order effects have been focused on since they can either complicate or contribute to the final fractionation depending on the way in which they are handled. Secondary-order phenomena represent all those perturbations leading to appreciable departures from the ideal retention theory. They are usually referred to as wall effects when they are due to the presence of a physical, interacting accumulation wall (21-26). These effects may have heavy consequences on the peak shape, leading to significant asymmetry. Therefore, one might wish a more accurate estimation of the higher-thanfirst-order moments. Great interest is arising in the complete peak shape analysis in SFFF because it can monitor the onset of complex processes such as the transition from steric to normal elution modes (23) and overloading effects and particle-wall interactions (25,26). If the sample concentration of an eluting band is too high (overloading), the normal (Gaussian) shape is skewed because the center of the band tends to move downstream more rapidly than either the front region or back region of the band itself. On the other hand, whether it is able to act as a short-range force or not, a simple, repulsive, particle-wall interaction (as the steric effect) might only give rise to a distortion in retention times rather than perturbation in the particle displacement velocity (25). Although these phenomena are often qualitatively detected by experts looking through eluted band parameters, their quantitative estimation from peak numerical analysis has yet to be described. To date, general theoretical treatment of an FFF process has been directed neither at obtaining statistical moments beyond the second one nor at deriving specific peak shape functions. However, this last point is not to be considered a serious drawback. In fact, general peak shape approximating functions have been reported that can be applied to a broad class of distribution functions (27). The crucial aspect to be pointed out is the question of whether or not the mathematical requirements of these approximating functions hold true under the specific circumstance of FFF. Furthermore, these hypotheses must be experimentally verified. As previously reported, there are general peak shape approximating functions: the Gram-Charlier series expansion of type A (GCA) (28, 29) and the Edgeworth-Cram&- asymptotic series expansion (EC) (28, 30-33). The GCA is a series expansion of a given frequency function f ( x ) with respect to the uth derivatives Z(’)(x)of the Gaussian function Z(x): y ( x ) = zc$qx)
~ ( x =) I/&
(1)
(2) x is the normalized variable x = ( t - m ) / u ,where m and u are the mean and the standard deviation of the peak, respectively. The uth derivative Z(’)(x)of the normal frequency can be computed as follows: exp(-x2/2)
Z(”(X)= H , ( x ) Z ( X ) / ( - ~ ) ’ ( ~ ~ / ~ ) (3) where H,(x) is the uth Hermite polynomial, which can be obtained by means of the recursive relation (34) H , + l b / f i ) = x f i H , ( x / f i ) - 2vH,l(x/fi) (4) with = 1 and H1(x/214 = ~ ( 2 l / ~ ) . It has been proven that, once developed to a given K term, the GCA series is a least-squares approximation of the original function f ( x ) (35). However, no general rules about the approximated function properties for the convergence of the CGA series have yet been defined, nor have high fitting degrees been attained with just a few GCA series expansion
121
terms. Thus, the EC series expansion appears much more useful than the other numerical methods mentioned above. The EC series are obtained from the CGA series by a partial ordering of the terms, according to the method defined by Edgeworth (30): ~ ( x= ) Y K ~ +) ? A x )
(5)
K
YKb)=
+ uCQA-2) =l
(6)
where y K ( x )is the EC series expansion developed up to the Kth term, Q,(-Z) indicates a linear aggregate in the derivatives of the normal frequency function Z(x) of maximum order 3u and contains rKcumulant coefficients of maximum order u, and rK(x) is the remainder. In the following two expressions (eqs 7 and €9,the first linear aggregation terms of the EC series are written in accordance with the general rule of building up Q, terms (36,Appendix B) by using multinomials and partition coefficients reported in Table 24.2 of ref 37:
(7) rI2 +-
(2 x
3)2
In ref 37 (expression 26.2.47),the full expressions for the third and fourth Q, terms are also reported. When the expressions for the Q,terms (eqs 7 and 8) are combined, EC series are thus expressed as products of the normal function Z(x) and Hermite polynomials with the cumulant coefficients rK as weighting factors. With the obtained aggregation term Q,(-Z), the EC series have the significant property of being asymptotically convergent for a broad class of frequency functions. It has been proved that chromatographic band migration along the column is asymptotically approximated by the EC series under infinite dilution conditions (38,39). The term “asymptotic” means, in practice, that the actual peak shape is well approximated by a series with a finite number of terms, provided the column is long enough (40). The EC series fitting is thus the improvement of the Gaussian approximation to the eluted peak. The basic theoretical property used in establishing the above-mentioned properties for the band shape evolution along the column is that the chromatographic process is stochastic with stationary and independent increments (39, 41, 42). In this work, it is supposed that the stochastic hypothesis holds true for an FFF band migration as well. As a matter of fact, this was the basic assumption when the general retention equation in FFF was derived (43):
R = (c*u)/(c*)(u)
(9)
where R is the retention ratio for a component subject to any form of FFF, c* is the equilibrium concentration, and u and ( u ) are the profile and average linear velocities of the fluid (44). In this concise expression, bracketed variables mean that particles actually homogeneously experience the entire velocity field within the channel. Therefore, a perfect equivalence between the mean space velocity (the average velocity computed over many particles a t the same time) and the mean time velocity (the average velocity computed over a single particle trajectory for a sufficient traveling time duration) must be assumed. This equivalence is attained through the random Brownian particle walk. This was also the basic model of the digital simulation for describing an SFFF elution mechanism (45). In this manner, the total residence time of the particle is a random variable, and it is proportional to the channel length. Therefore, the process can easily be accepted to having the same stochastic properties as a linear chromatographic
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
event. The latter assumption defines a FFF experiment as a stochastic process with linear independent increments (41). Under these hypotheses, the EC series approximation for an FFF band migration is more than a numerical fitting tool since it displays a theoretical background. The work discussed in the present paper deals with the peak shape analysis of SFFF experiments by EC series expansion. A comparison between different estimation methods of peak parameter values (mean, standard deviation) is also taken into account. Numeric integration, graphical approach, and EC series are checked with respect to their ability to monitor the effect changing experimental conditions (particle diameter, field strength, sample injection loading) has on the peak shape. The main purpose of this work is to test the EC series’ ability to perceive the onset of nonlinearity conditions. Moreover, once linearity aspects have been verified, the ability of FFF to determine higher-than-second-orderpeak parameters, and thus the nature and entity of secondary-order effects, is emphasized.
THEORY EC Series Peak Shape Fitting. The experimental peak shape, which is the output response Ysp as a function of time t , is related to a standardized frequency function f ( x ) with unit area as y,,(t) = f(x)A/u
(10)
where u is the peak standard deviation. The peak area ( A ) and the normalized time variable ( w ) are both defined as follows:
(11) where m is the peak mean. A theoretically computed output response Y,,, can be carried out by an EC series development up to the Kth term:
(4) The number of nodes in the remainder R K ( x ) ,approximated by AK,,(eq 15), increases as the K grade is raised and is almost symmetrical with respect to x = 0. The simultaneous consistency with the general features mentioned above is a necessary condition for the process to be classified as operating under linear conditions. When, as the quantity injected is increased, these conditions no longer hold true, it is usually indication of the onset of nonlinearity effects. This last point has been proven in different systems such as GC packed and capillary columns and ion chromatography (47-50). An appealing perspective of such a peak shape analysis lies in its “intrinsic” ability to monitor nonlinearity effects through peak parameters of a given eluted band. As far as sample overloading is concerned, one might bypass a simplistic-but hard to manage-comparison of peak profiles obtained by injecting different sample amounts in order to see the onset of any nonlinearity effects. Obviously, such a powerful approach must be properly verified, particularly with reference to any specific applications. Peak Parameters. Among the several peak parameters that are outputs of an EC fitting procedure, the first two cumulant coefficients rl and r2are most usually referred to as peak shape parameters (36):
rl = s r2 = E
(17)
x,= -(1/2)S
(19)
(18) They are called skewness and excess, and respectively, they express the degree of skewness and “flatness” with respect to the Gaussian reference shape. Some other peak attributes are directly obtained from normalized peak shapes:
X,is Charlier’s measure of skewness, most properly defined as the mode normalized coordinate (28). This parameter represents the difference between peak maximum tR and peak barycentre m. Hence, in time units, one can obtain It, = xou
where y K ( x )is given by eq 6. The remainder term R&) is formally defined as
RK = (A/g)r&)
Therefore, the retention time error generated by using peak maxima might be defined as a function of the peak skewness. In fact, by combining eq 19 and eq 20, one gets (ltR/tR)%
where the term RK is (14)
r&) being defined in eq 5. The nonlinear least-squares fitting is defined as with respect to parameters A , u, m, rl,and rK. Series expansion improves the degree of fitting up to a maximum value, depending on the peak shape and noise. It has been numerically proven that the EC series approximates experimental skewed peaks, under linear conditions, within a skewness ( S ) range of 0 < S < 0.8 (46). The goodness of fit is defined by four rules (46): (1)The maximum expansion order K is related, by empirical rules, to the degree of skewness and to the noise. (2) The parameter set ( A , m, u , ..., rK)K for a given expansion remains almost stable as K is increased to its maximum value (see rule 1). (3) The mean approximation error, expressed as
where N , and np are respectively the number of points and the number of parameters and Y,, is the peak height, lowers as K is increased to a value consistent with the intrinsic noise level.
(20)
= 100(1/2)SCT/tR
(21)
Skewness values might handily be evaluated through a graphically achievable quantity such as the peak asymmetry A,, which is usually defined as the ratio between the peak tailing section and peak fronting section a t 10% of the peak height (51). Since the approximate relation S=A,-l
(22)
most likely holds true for an SFFF peak, as previously observed in other chromatographic systems for low asymmetry values (ca. A , < 1.4) (46),one can rearrange eq 21 by using the plate height expression
H = L/(tR2/2)
(23)
where L is the channel length. Thus, one obtains an expression of the retention time error as a function of graphically available quantities: ( A t R / t R ) % = 100((H/L)(AS/2 - 0.5)2)’/2
(24)
where the exact plate height H value expressed in eq 23 is approximated by means of a conventional, graphical evaluation of u from the recorded trace on paper. One of the simplest interpretations of moderately skewed peaks might be depicted as a convolution of a “pure” Gaussian shape, with a standard deviation UG and an exponential decay function with time constant T (9). Hence, the total variance
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
of the convoluted peak can be obtained as in (49)
assuming that the value of y is independent of d (23),the expression of the inversion point in SFFF is expressed as
The exponential time constant T can be related to the total standard deviation and skewness (48, 49) as 7
= atot(S/2)’/3 =
aG/atot
(27)
R = 36kT/xd3(Ap)Gw+ 3 y d / w
(28)
where co is the wall concentration, x the applied field axis ( x = 0 a t the wall), and 1 the mean solute layer thickness. With the concentration profile so specified and the longitudinal flow known to be parabolic, the retention ratio R = V,/ Vo,where V , and Vo are the retention volume and the void volume, respectively, can be derived as
R = 6h[coth ( 1 / 2 h ) - 2x1
(29)
where h = l / w and w is the channel thickness. At small X values, the retention ratio can be approximated as
R = 6X
(30) Since useful retention in SFFF is usually maintained at or less than R = 0.3, this simplified expression is satisfactory for determining sample parameters from simple retention time measurements in most situations (6). In fact, in SFFF, the expression for X becomes
X = 6 k T / T (Ap)w Gd3 (31) where k is the Boltzmann constant, 5” the temperature, Ap the difference in density between particle and carrier, G the gravitational acceleration, and d the particle diameter. From eq 31, it is evident that larger particles interact most strongly with the applied field and thus they are retained longer than smaller particles. However, as the solute diameter becomes appreciable in comparison to its own mean layer thickness 1, the retention mechanism gradually undergoes a transition to a steric elution mode (23). In steric SFFF, retention is largely determined by particle diameter. The retention order displays a trend that is just opposite that of the normal elution mode. As an appreciable steric effect becomes operative in SFFF, a mixed elution process offsets the application of eq 30, the expression of the retention ratio R being approximated as (23) R = 6h 6 y a (32)
+
The steric factor y is a dimensionless parameter of order unity, and the term CY is given by (Y
= d/2w
(34)
(26)
it can be directly evaluated by substituting eq 26 into eq 25. The peak fidelity ratio (eq 27) is actually an easy way to measure those detectable extracolumn band-broadening effects referred to as affecting the peak shape as exponential decay functions do (49). Retention and Steric Effects in SFFF. In normal (or “Brownian”) SFFF, the sample is forced toward the wall of the channel by the imposed field to form a layer of uniform thickness, such a thickness depending on the interaction between the sample and the field as well as the opposing Brownian motion (19,431. Thus, the sample layer settles to a steady-state exponential distribution: c = co exp(-x/l)
di = ( 3 6 k T / ~h(p ) ~ G ) ’ / ~
where di is the diameter of the particles eluting at the inversion point. For still larger particles, the separation mechanism turns into a “full” steric mode. By substituting eqs 30, 31, and 33 into eq 32, one gets
If we define the “peak fidelity ratio” ( 9 ) as
a
123
(33)
where d is the particle diameter and w the channel thickness. As expressed in eq 32, the retention volume increases with particle diameter until steric effects dominate. The minimum in retention R at which transition occurs is obtained by applying the condition dR/dd = 0 to eq 32. By
(35)
Just as the first term accounts for the normal SFFF retention mechanism, the second term accounts for the steric elution mode. The inverse cubic dependence of the retention ratio R on the particle diameter d for normal SFFF (first term of eq 35) is thus counterbalanced by a direct relationship between R and d (second term in eq 35). Therefore, according to this “corrected” retention expression, whenever a steric contribution starts to emerge, one might expect the corresponding experimental R values to increase with respect to those predicted on the basis of the “standard” retention theory alone (eqs 30 and 31). This is the most common experimental evidence of the onset of a steric effect. However, an accurate “a priori” evaluation of the steric correction coefficient on retention cannot escape the complex dependence of the dimensionless factor y on lift forces and related hydrodynamic effects (53). Experimental evidence that y is also a function of particle diameter has already been reported (53-55). Therefore, derivation of an expression for steric transition cannot be rigorously performed. The onset of steric effects and their influence on particle parameters (size and density) is thus a problem that has not yet been fully worked out. However, besides the transition region from normal mode to steric mode, mixed elution mechanism acting together can turn out strong peak deformation and tailing. These features have already been discussed in the literature for chromatographic mixed retention processes (39,52).Thus, peak shape analysis might help in blunting this theoretical drawback since it might correlate accurately determined peak shape parameters and the degree of steric contribution on retention. EXPERIMENTAL SECTION The sedimentation FFF apparatus used for this study was the Model SlOl SdFFF from FFFractionation, Inc. (Salt Lake City, UT). The outer wall material was bare polished HastelloyC, whose nominal composition is 59% Ni, 16% Cr, 16% Mo, 5% Fe, and 4% W with traces of Mn and Si. The spacer material used in the channel consisted of a nominal 0.0254-cm-thick Mylar strip. The ribbonlike channel cut from this strip was 89.3 cm in length and 2 cm in width. The channel void volume was 4.89 mL as determined by the measured retention volume of a nonretained probe. Channel flow was generated by a Spectra-Physics HPLC column Model SP8700. The outlet tube of the system was connected to an UVIDEC 100 variable-wavelengthUV detector operating at 254 nm (Jasco Ltd., Japan) and whose signal was fed to a SPEEDOMAX recorder (Leeds & Northrup, Sumneytown Pike, North Wales, PA). Data handling was performed by feeding the detector analogic output to an ACRO-900 12-bit 1 / 0 data acquisition system (Acrosystems Co., Beverly, MS). The 1/0 board was connected to an Olivetti M24 DOS personal computer (Olivetti Co., Italy). Both the 1/0digital board setup menu and fast Fourier transform (FFT) digital data filtering were driven by the LABTECH NOTEBOOK software (Laboratory Technologies Corporation, Wilmington, MA). The sample consisted of Dow monodispersed, polystyrene latex beads obtained from Serva Feinbiochemica, Heidelberg, FRG, with nominal diameters d = 0.481 f 0.0018 and 0.624 f 0.0052 wm and density 6 = 1.05 g/cm3. Particle concentration was diluted to 1%(w/w) and 0.1% (w/w), and sample loading was 0.18 and 0.011 mg, respectively. The carrier solution was prepared by using Milli-Q deionized water (Millipore Corporation, Bedford, MA), and helium was bubbled through it to eliminate air. The solution was made by adding 0.01% of the detergent FL-70(Fisher Scientific, Fair Lawn,
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
Table I . Onset of Steric Effect: Peak Parameter Patterns Obtained by Nonlinear Least-Squares Fitting at Different EC Series ExDansion Orders K and Different Field Strengths' field, g
K
H , cm
0
0.37 0.37 0.41 0.44 0.43 0.43 0.43 0.43
S
E
CV%
0.64 0.69 0.66 0.63 0.63 0.64
2.5 1.2 1.0 0.2 0.2 0.4 0.4 0.5
61
Table 11. Onset of Steric Effect: Peak Parameter Patterns Obtained by Nonlinear Least-Squares Fitting at Different EC Series Expansion Orders K and Different Field Strengthsn field, g
K
H, cm
S
0 1 2 3 4 5 6
0.26 0.27 0.27 0.28 0.29 0.29 0.29 0.29
0.32 0.32 0.38 0.30 0.34 0.34 0.25
0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.18
0.59 0.58 0.54 0.61 0.60 0.59 0.59
0.13 0.15 0.15 0.16 0.16 0.16 0.16 0.16
0.81 0.80 0.70 0.70 0.78 0.82 0.84
0.19 0.22 0.22 0.24 0.26 0.25 0.25
0.96 0.99 0.84 0.54 0.36 0.10
E
CV %
0.21 0.43 0.39 0.62 0.63 0.76
2.2 1.o 1.0 0.8 0.8 0.5 0.6 1.o
0.50 0.41 0.46 0.44 0.43 0.43
3.8 0.8 0.7 0.5 0.5 0.5 0.6 0.6
0.98 0.97 0.87 0.89 0.89 0.89
5.0 1.3 1.1 0.6 0.8 0.6 0.8 1.6
1.21 1.20 0.49 0.17 0.32
7.1 3.1 2.4 2.1 1.6 1.9 5.9
42 1 2 3 4 J
6 r
I
0.36 0.35 0.18 0.24 0.30 0.29 0.30
83
I
61 0 1 2 3 4 5 6 I
0.25 0.25 0.25 0.25 0.26 0.26 0.26 0.26
0.48 0.48 0.37 0.30 0.34 0.39 0.38
0.30 0.21 0.12 0.23 0.22 0.19
3.1 0.9 0.8 0.6 0.3 0.4 0.4 0.5
108
0 1 2 3 4 5 6 I
83 0 1 2 3 4 3
-
6 I
0.18 0.19 0.18 0.18 0.18 0.19 0.19 0.19
0.75 0.81 0.70 0.50 0.48 0.43 0.35
0.50 0.31 0.28 0.44 0.38 0.40
5.1 2.6 1.3 1.3 0.7 0.8 1.3 1.9
0.80 0.58 0.33 0.83 0.67 0.87
7.0 3.9 2.2 2.2 1.6 1.4 2.6 3.8
0.97 0.63 0.81 0.50 0.31 0.32
7.6 3.9 2.5 2.2 2.2 2.1 3.2 3.6
136
0 1 2 3 4
5 6
108
0 1 2 3 4
5 6 I
0.20 0.21 0.21 0.20 0.21 0.22 0.22 0.22
0.85 0.90 0.82 0.58 0.55 0.43 0.37
0.17 0.19 0.18 0.18 0.18 0.18 0.18 0.18
0.95 0.97 0.84 0.82 0.68 0.56 0.45
168 0 1 2 3 4 5 6 I
Sample: 0.480 pm; 0.011 mg N J ) and 0.002% sodium azide. A flow rate of 1.0 mL/min was used in all the experiments.
COMPUTATION SECTION The experimental fractogram, from a digital viewpoint, was a sequence of equally spaced (0.03 Hz)Y, data. I t was then transformed to a frequency domain by a fast Fourier transform (FFT) algorithm (56). The frequency spectrum was then filtered by applying a low-pass filtering function whose frequency cutoff was determined in order to accurately keep the original peak shapes. Since the forward Fourier transform of a Gaussian-like profile is still a Gaussian-like function rapidly tending toward zero ( 5 6 ) ,the frequency cutoff was graphically determined as the point where the standard deviations of the power spectrum amplitudes remained constant. At this cutoff point, the power spectrum dropped to less than -0.03 of its maximum value (50, 5 7 ) . The filtered peak, in the time domain, was afterward restored by backward Fourier transform. Both the base line and linear drift were subtracted by linear interpolation of those points flanking the peak maximum by more than &5n. Standard deviation and peak mean starting values for the minimization procedures were
0 1 2 3 4
5 6 7
"Sample: 0.624 pm; 0.011 mg. determined by intergration methods, as described elsewhere (36). The nonlinear least-squares fitting was initiated by using the zeroth-order EC series term, which actually is the Gaussian function. The fitting procedure was then carried on by increasing the series expansion order K until (CV%), was significantly lower than any former (CV%)K-lvalue. The fitting patterns are reported in Table I and Table 11. The EC series least-squares fitting routine was a FORTRAN compiled software run on an IBM System/2 Model 70 386 (International Business Machine Corp.) DOS personal computer. The general minimizing routine (58) integrated into the EC series package required the minimized function FK values, and its gradients referred to different parameters (59). They can be expressed as
(36) where p , represents the parameters A , m, u , r,, ..., r K . The EC series least-squares fitting routine also includes the computation of first peak momenta ( m , u2) by numerical integration on discrete sets of digitized peak data. These values were used to derive the retention and plate height data ( R , H) reported in Table I11 and Table IV and referred to as integration data. As far as graphical analysis is concerned, peak asymmetry A , (51),retention time corresponding to the peak maximum tR, and peak standard deviation u were simply measured on paper records, and the quantities derived are shown in Table I11 and Table IV.
ANALYTICAL CHEMISTRY, VOL. 63,NO. 2, JANUARY 15, 1991
125
- , 0
3
2
1
'
K '
Flgwe 2. Relative approximation error (CV%)as a function of the EC series expansion order (K), 0.624-pm polystyrene latex beads, 42 gravities (500 rpm): (0)no overloading (0.011 mg); (+) overloading (0.18 mg). 1
0.1 0.7 0.6 0.5
-5
-3
-1
1 c
-
I
x 4 t - m)/o
Flgure 1. EC fifth-order approximation, 0.624-pm polystyrene latex beads, 42 gravities (500 rpm): (-) experimental peak: (+++)fitted peak. (a) Injected quantity = 0.01 1 mg (no overloading): (b) injected
quantity = 0.18 mg (overloading).
RESULTS AND DISCUSSION In the present study, three experimental, independent variables affecting peak shape have been considered: (1) gravitational field strength (27-168 gravities (g)); (2) particle size (0.480 and 0.624 pm); and (3) injected sample loading (0.011 and 0.18 mg). An experimental design was set up in order to study the individual factors described above and their influence on peak parameters. A moderately low carrier ionic strength (3 X 10"' M sodium azide) was chosen in order to enhance the onset of nonlinearity sample overloading (25). According to this experimental strategy, only two kinds of secondary-order effects were considered: nonlinearity overloading at low carrier ionic strength (25) and transition from the normal mode toward the steric mode (23). Actually these are the two that have most often been described in the literature as affecting SFFF. Sample overloading effects on the peak shape are shown in Figure 1 where two peak profiles obtained under all the same experimental conditions except injected sample loadings are reported. Injection plug amounts where chosen, first, as a value for which the carrier solution employed gave rise to a detectable overloading (0.18 mg) and, second, as a value having a negligible effect on retention (0.011 mg) (25). In the same plots, the best EC series fitting functions are superimposed on their input experimental peaks. As far as peak shape is concerned, an SFFF overloading looks quite similar to a nonlinear Langmuir-type effect in chromatography (60). The peak shape is tailed and triangular; its fitting profile shows a typical dip fronting on the peak because of the rippling nature of the series expansion terms at high skewness values. The approximation patterns obtained by ever increasing the EC series expansion order for the peaks reported in Figure 1are shown in Figure 2 where the dependence of CV% values on the EC series expansion order K is plotted. A minimum
3
I
0
K 5
Figure 3. Skewness (S) versus EC series expansion order (K), 0.624-pm polystyrene latex beads, 42 gravities (500 rpm): (0)no overloading (0.011 mg): (+) overloading (0.18 mg).
E
0.s
0.1
-
..
0
t
2
3
'
K 5
Figure 4. Excess ( E )versus EC series expansion order ( K ) , 0.624-pm polystyrene latex beads, 42 gravities (500 rpm): (0)no overloading (0.011 mg); (+) overloading (0.18 mg).
is attained (CV% = 0.5) upon increasing the expansion order for both of the analyzed profiles; once the best CV% has been reached, the approximation degree always gets worse, no matter how much further the expansion order increases. This common behavior indicates that the higher order added EC series terms have become comparable to the experimental error (46). The dependence of skewness and excess values on the expansion order is shown in Figure 3 and Figure 4, respectively. Peak parameters are stable and meaningful only for the fitting procedure run on a nonoverloaded profile. Moreover, the fitting pattern for an overloaded peak exhibits a nonhomogeneous approximation degree over the entire profile. Furthermore, it shows a lower number of clearly defined nodes
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
126
Table 111. Comparison between Fitting and Graphical Analysis: Onset of Steric Effect" EC peak parameters
integration
no.
field, g
Hf
K
S
Hf
R
Rex,
dexpb"
1A 2A 3A 4A 5A
61 83 108 136 168 42 61 83 108
0.44 0.26 0.19 0.22 0.18 0.29 0.18 0.16 0.26
3 4 4 5 3
0.18 0.30
0.46 0.26 0.21 0.23 0.29 0.31 0.19 0.18 0.38
0.060
0.060 0.047 0.039 0.034 0.031 0.057 0.045 0.037 0.032
0.467 0.451 0.437 0.431 0.414 0.559 0.535 0.514 0.497
1B 2B 3B 4B
0.50 0.55 0.84 0.34 0.61 0.70 0.54
5 4 3 4
0.047 0.038 0.033 0.030 0.057 0.044
0.036 0.031
graphical analysis lrJ l/d didbf 2.54 1.99 1.65 1.44 1.31 2.41 1.91 1.57 1.35
5.3 4.1 3.4 3.0 2.7 3.9 3.1 2.5 2.2
(d/di)%
1.10 1.00 0.97 0.91 0.87 1.20 1.10 1.00 0.97
+%e
44 48 49 53 55 52 57 62 64
98.80 84.70 78.70 63.20 66.30 83.00 79.95 70.95
Experimental sets: 1A-5A = 0.480 pm; 1B-4B = 0.624 pm. Values obtained by eq 31. Values obtained by eq 30. Values calculated by eq 34. e Values calculated by eqs 25-27. f H is expressed in cm; 1 is expressed in pm; d , and der, are expressed in pm.
Table IV. Comparison between Fitting, Integration, and Graphical Analysis: Onset of Steric Effect Peak Distortion and Retention Error" EC peak parameters no.
field, g
Hf
S
Xob
A~R'
(AtR/tR)YCc
ltRf
1A 2A 3A 4A 5A
61 83 108 136 168 42 61 83 108
0.44 0.26 0.19 0.22 0.18 0.29 0.18 0.16 0.26
0.18 0.30 0.52
-0.09 -0.15 -0.26 -0.28 -0.42 -0.17 -0.31 -0.35 -0.27
-30
0.6
-50
0.8 1.2 1.3 1.9 1.0 1.4 1.5 1.5
-52 -74 -160 -261 -254 -61 -78 -121 -229
1B 2B 3B 4B
0.55 0.84 0.34 0.61
0.70 0.54
-91 -121 -183
-50 -91 -119 -135
integration ( A ~ R / ~ R ) % A,d 1.0 1.2 2.1 2.9 2.6 1.2 1.2 1.5 2.5
graphical analysis
Hf
(AtR/tR)%e
1.3 1.7
0.46 0.26
1.8
1.4
0.31
1.2
1.1
Experimental sets: 1A-5A = 0.480 pm; 1B-4B = 0.624 pm. bValues obtained by eq 19. Values obtained by eq 21. dAsymmetry values expressed as in eq 22. eValues calculated by eq 24. f H is expressed in cm; AtR in s.
-5
- 3
-1
1
3
x
5
x
3
& & + -32 -1
Figure 5. Residuals of EC series fitting, fifthader expansion, 0.624-pm polystyrene latex beads, 42 gravities (500 rpm). D% = percent differences between fitted and experimental peak; (a) no overloading (0.01 1 mg); (b) overloading (0.18 mg).
and less symmetrical ripples (see Figure 5) if the percent differences between experimental and fitted peaks are compared for both of the chosen sample loading conditions. Once tested for the general "goodness of fitting" rules previously described in the Theory section, these features show this procedure is able to unambiguously discriminate between overloading and nonoverloading conditions in SFFF. This EC series discriminating ability has also been reported for GC and ion chromatography (48-50). Let us now consider how the progressive onset of steric effects is monitored by the present EC series peak shape analysis. In Figure 6, the experimental peaks and relative best fitting functions are compared for a 0.480-~mpolystyrene latex sample run a t different field strengths (from 61 gravities to
i
-I
l
l
l
4
l
-1
l
i
0
l
l
1
l
l
4
l
i
6
X Figure 6. Steric effect onset: Normalized experimental (-) and EC fitted peaks (symbols) versus normalized time variable ( X ) at different field strengths, 0.480-pm latex polystyrene beads. Injected quantify = 0.01 1 mg; (A)61 gravities (600 rpm), ( X ) 83 gravities (700 rpm), (V)108 gravities (800 rpm), (+) 136 gravities (900 rpm), ( 0 )168 gravities (1000 rpm).
168 gravities: 600, 700,800, 900, and lo00 rpm, respectively). The sample amount injected (0.011 mg) was chosen in order to reasonably avoid an appreciable overloading even a t the highest field strengths set for these experiments (25). The onset of a progressive peak leaning is detected by a mode point shift-i.e., the abscissa of the peak maximum-to negative values of the normalized time variable. Therefore, peak shape
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
127
1.4
1.3 1.1 10.9
-
0.1 0.7 0.8 0.5 0.4 0.3 0.2 0.1
I
I I
-I
I
-4
I
-2
I
I
I
I
2
0
I
4
-
-
0
,
I
X Figure 7. EC thirdsrder approximation: single-peak components, 0.48O-fim latex polystyrene beads, 61 gravities (600 rpm), injected quanti = 0.01 1 mg; (-) experimental peak, (+++)fitted peak, Z(x) = Gaussian component, 1 = O , ( - Z ) , 2 = O,(-Z), 3 = 03(-Z).
cv%:i,
0
2 1
2
3
4
5
I
7 K 0
Flgure @. Skewness (S) versus EC series expansion order (K): steric effect onset, 0.480-pm polystyrene beads, injected quantity = 0.01 1 mg; (0) 61 gravities (600 rpm), (+) 83 gravities (700 rpm), (0)108 gravities (800rpm), (A)136 gravities (900 rpm), (X) 168 gravities (1000 rpm).
10
2
E ::: 1 --
7
1.7 1.1 1.5 1.4 1.3 1.2 1.1 10.9 0.1 0.7
-
0.8
-
0.2 0.5 0.4
0.3 0
1
2
3
4
5
6
K6
Flgure E. Relative approximation error (CV%)as a function of the EC series expansion order ( K ) : steric effect onset, 0.480-pm latex polystyrene beads, injected quantity = 0.011 mg; (0)61 gravities (600 rpm), (+) 83 gravities (700 rpm), (0)108 gravities (800 rpm), (A)136 gravities (900 rpm), (X) 168 gravities (1000 rpm).
distortion might qualitatively indicate the onset of a steric effect. Nevertheless, quantitative evaluation of this steric contribution on peak parameters is possible only by using EC series numerical fitting. In fact, such EC series fitting experiments prove highly accurate in monitoring any peak shape difference, even for almost Gaussian profiles. One can also note a dip front on the most skewed peaks, as observed above for overloading (Figure lb). In Figure 7, the best EC curve fit of the less skewed peak obtained at the weakest field strength is shown together with its own single peak components: a slight but appreciable peak shape distortion is evident once the experimental peak has been compared to its normal component. The nature of a normal-to-steric elution mode onset can be analyzed through the fitting pattern as was previously performed for the peak shape asymmetry due to overloading. As shown in Table I, the general peak shape is always sufficiently approximated by EC series expanded up to the fourth to fifth order. The fitting accuracy spans no more than fl%, even for tailed peak shapes, although fitting patterns and relative CV% values (also shown in Figure 8) indicate that the EC series lacks fitting ability when applied to the most highly skewed peaks (S > 0.8). These intrinsic limits have previously been discussed in the literature (46).Table I1 also reports the fitting patterns obtained for 0.624-pm latex particles a t different field strengths. When comparing Table I and Table 11, one can observe no significant difference in the pattern fitting behavior for either particle diameter. Therefore, with respect to the extent of EC series approximation,
0.1
,
0 ’ 0
Y
1
2
3
4
5
I
K ’
Figure 10. Excess ( E )versus EC series expansion order ( K ) : sterlc effect onset, 0.480-pm polystyrene beads, injected quantity = 0.01 1 mg; (0)61 gravities (600 rpm), (+) 83 gravities (700 rpm), (0) 108 gravities (BOO rpm), (A)136 gravities (900 rpm), (X) 168 gravities (1000 rpm).
the effect of overloading and the onset of steric behavior are similar; Le., a t comparable peak asymmetries they reach their own best approximating degrees in correspondence with the same expansion orders. Nonetheless, peak parameter sets obtained by increasing the EC series expansion order actually appear more stable for the onset of steric transition rather than for nonlinear overloading effect. This latter feature can be focused on by comparing Figure 9 and Figure 3 for skewness as well as Figure 10 and Figure 4 for excess patterns. This finding might confirm the predicted difference in the effect on the peak shape distortion exerted by two secondary-order phenomena derived from interactions acting as particle-particle repulsion (overloading) or particle-wall repulsion (steric effect) (25). These peak shape parameter properties seem to belong more strictly to the general necessary conditions for a “linear nonideal” process. In fact, one might picture the total particle retention as a mixed event due to two convoluting processes: the normal and steric elution modes. Actually, they act on the total retention in opposite ways (23). A precise tractability of such a complex probabilistic dynamic process stands beyond the aim of the present paper, although it can be seen from these EC least-squares approximation studies that the onset of steric foldback in SFFF appears to belong to a more general class of nonideal processes. Nevertheless, such an argument should be more extensively supported by a proper study of polydisperse samples with specific experimental design. A rigorous check
128
ANALYTICAL CHEMISTRY, VOL. 63, NO. 2, JANUARY 15, 1991
of the sample dispersity should be performed since an intrinsic, non-Gaussian polydispersity could contribute to peak distortion. In fact, with an increase in field strength, the raised selectivity on sets of particles that could be only slightly different in diameter might turn out a peak distortion if the sample polydispersity is non-Gaussian shaped. Otherwise, a steric onset might be depicted as an effect of the nonhomogeneous selectivity on normal distributed polydispersity since, with an increase in field strength, the enhanced selectivity rapidly drops because larger particles elute earlier than predicted. In Table 111, two sets of retention parameter values derived from calculation based on graphical analysis, integration, and numerical fitting are reported. Integration values are the input conditions for the fitting procedure (expansion order K = 0) and were calculated as previously described in the Computation Section. The onset of a normal-toward-steric elution mode is clearly indicated by a progressive underestimation of the experimental particle diameter (de.+,). This is calculated from experimental retention ratios (Rerpfrom the retention volume at the peak summit, eq 30) by using the “standard” relation for retention (eq 31) and by increasing the field strength, as suggested in previous literature (61) and described above in the Theory section. Such an experimental evaluation of the particle diameter (de,?) is obviously “incorrect” when steric effects offset the application of eq 31. As a matter of fact, this dexpparameter does not take into account the second term of eq 35 (e.g., the steric correlation on normal retention), which is most properly the relation for mixed normal-steric retention. However, an accurate sizing of the samples is not to be intended by means of the d e x pparameter. On the contrary, d e x pvalues can actually be used to monitor this steric bias on retention. From the data reported in Table 111, one can distinguish for both samples examined (0.480 pm, 2A-5A set; 0.624 pm* 1B-4B set) the onset of steric effects, even a t mean solute layer thicknesses (2) 4 times larger than the corresponding nominal particle diameter ( d ) . Table I11 also reports the steric inversion diameters (d,)calculated by eq 34 with an approximate assumption of y = 1. The lei el of steric contribution reached by varying the field strength might be referred to as a “nominal diameter over inversion diameter” ratio ((did,) %). The experimental conditions chosen in the present work span ( d / d , ) % values from 240% to ~ 6 0 % Although . this range clearly stands fairly far from the actual steric transition point, peak shape analysis is distinctly able to detect a significant steric contribution in the normal retention mode. The sensitivity of an EC series approach to stress such an effect might be focused on observing (Table 111) the ( d / d , ) % value limit (60%) beyond which EC series fittings break down. One can observe that this boundary corresponds to a particle diameter estimation bias, expressed as the nominal diameter over experimental diameter ratio ( ( d / d e x p ) % )that , is not higher than 20%. Yet these EC series applicability limits on the weak steric contribution must be considered not as an intrinsic drawback but actually as an appealing approach since one needs to more closely focus on the moderate steric contribution on retention. Moreover, the data shown in Table I11 could also display an interesting, slightly perceptible correlation between peak shape markers (skewness) and (did,) % values. This observation might suggest that the EC series shape analysis might be quantitatively applied to the analysis of the steric bias on a retention measurement. A full analysis of this relationship would require a more extended experimental study and extends quite beyond the purpose of the present work. Consistency between integration and the numerical EC peak shape fitting analysis is also monitored in Table IV by comparing the differences in retention times between fitted peak
maxima and normalized time variable origins (AtR) as well as making a comparison between the retention time error (( I t R / t R ) % ) reached by using the peak maximum rather than by more properly using the peak mean abscissa. These values obtained by integration procedures are consistent with those calculated from skewness values by EC series fitting through the mode normalized coordinate expression suggested by (28) (eq 19). Significant discrepancies are obC r a m 5 (X,) served only for those peaks obtained a t the highest fields, thus once again confirming the EC series fitting limits for heavily skewed (5’ > 0.8) peaks. As far as the benefits derived from the numerical fitting procedure are concerned, one might wonder whether such peak shape control is to be considered a necessary requirement whenever retention parameters are to be determined from peaks recorded on paper as is often the case. In order to define a relatively handy test with which to check peak shape complexions, besides using a complete peak shape fitting procedure, we can apply the simple relation between the graphically achievable peak asymmetry (A,) and relative skewness (eq 22) which holds true for ca. A, < 1.4. Thus, using eq 22, one can evaluate the retention time error ( ( I t R / t R %). ) These simple graphical checks might suggest whether further, accurate peak shape numerical fitting is necessary in order to attain information on peak parameters. In fact, only for highly symmetrical peaks can one assume that no steric contribution affects the retention, and thus, a direct, accurate particle diameter determination becomes possible. Table IV compares these ( A t R / t R ) % values to those obtained by integration and EC fitting. Whenever peak asymmetry (A,)is lower than 1.3-1.4, one can assume that an almost normal elution mode prevails under these elution conditions and a maximum bias of = L O % on retention is originated by using the peak maximum. No graphical approach is available for the highest skewed peak (for values of asymmetry A, > 1.4, eq 22 is no longer reliable), and a maximum error lower than 3% is observed. Therefore the highest bias on retention (see Table 111) is due to the steric effect on band migration. Obviously, all these observations hold true only for sharply monodispersed samples. Therefore, whenever a peak distortion is observed, a coupled study of both peak skewness and plate height dependence on field strength might indicate that this finding can be ascribed to polydispersity or to steric effects. Final focusing on the highest symmetric peak reveals once again (Table 111) a skewness value as low as 0.18. If this skewness effect is ascribable to an exponential, extracolumn decay factor (eq 26), a high peak fidelity ratio (a) (9,49) (eq 27) is obtained. This indicates a very good overall instrumental setup as well as a correct injection and relaxation procedure. Therefore, in these sets of experiments, no peak asymmetry can be ascribed to instrumentation or experimental procedure. As far as a comparison between peak parameter determination methods is concerned, it has already been widely demonstrated (62, 63) that large systematic error5 appear in the computation of the peak moments by numerical integration. Analysis of chromatographic peaks by the EC series has shown good agreement between integration and numerical fitting only for the first two peak parameters ( m , U ) (36). Therefore, only standard deviations, expressed as the plate height H , are compared in Table 111. No significant differences between plate height measurements based on integration and the EC fitting procedure are reported, except for the highest fields where the EC series shows its limitations (see data 5A and 4B).
CONCLUSION The first peak shape approach to an F F F process by the EC series least-squares fitting method reveals several features
ANALYTICAL CHEMISTRY, VOL.
common to a chromatographic process. For accurate determinations of peak parameters and secondary-order effects as well as for fine instrumental tuning, peak shape control must be accepted as a necessary tool, even if performed by simple graphical methods. Moreover, once the most proper peak shape for sharply monodisperse samples has been established, other important tasks in current SFFF practice (i.e., polydispersity determination) can be considered by means of numerical peak shape fitting. However, a full theoretical description of the quantitative dependence of those peak parameters obtainable by EC series analysis (S, E ) on the onset of secondary-order phenomena (steric effect, sample overloading) has yet to be worked out. Likewise, interesting effects such as particle-particle interaction, particle-wall interaction, and other secondary-order phenomena acting on retention and peak shapes could be more accurately estimated by means of a properly designed experimental approach in order to more clearly discriminate between different nonidealities. Such an advanced study, which indeed has not been examined in this work, would benefit from the EC series' powerful approach.
ACKNOWLEDGMENT We thank J. C. Giddings, FFF Research Center, University of Utah, for having made available an SFFF prototype and M. N. Myers, FFF Research Center, University of Utah, for having set up the system and supervised its technical main-
GLOSSARY peak area peak asymmetry (51) (eq 22) sample concentration across the wall equilibrium concentration of the sample (44) (eq 9) sample concentration a t the accumulation wall percent coefficient of variation (eq 16) nominal particle diameter experimental diameter diameter of steric inversion (23) (eq 34) peak excess Edgeworth-CramBr asymptotic series expansion (28, 30-33) (eq 6 ) minimized function (eq 36) fast Fourier transform standardized frequency function field strength in gravities gravitational acceleration (cm/s2) Gram-Charlier series expansion of type A (28,29) (eq 1) Hermite polynomial of order u (34) (eqs 3 and 4) plate height Bolztmann constant order of series expansion solute mean layer thickness channel length peak mean number of parameters for a minimization pattern (eq 16) number of points for a minimization pattern (eq 16) peak shape parameters EC series terms of order v (eq 6) retention ratio EC series remainder (eq 5, eq 14) peak skewness retention time absolute temperature velocity profile of the fluid (44) (eq 9) average linear velocity of the fluid (44) (eq 9) void volume retention volume channel thickness
63,NO. 2, JANUARY 15, 1991
129
mode normalized coordinate (28) (eq 19) computed peak profile by EC series development (eq 12) digitized peak profile peak height (eq 16) EC series expansion of order K experimental peak profile Gaussian function (eqs 1, 2, and 6) retention parameter in steric elution mode (23) (eq 30) sample density minimized function gradient (eq 36) approximated expression of the remainder R K (eq 15) density difference between sample and carrier difference between peak maximum and peak barycentre in time units (eq 20) peak fidelity ratio (9) (eq 27) dimensionless parameter of the steric contribution (23) (eq 32) EC series cumulant coefficients dimensionless solute mean layer thickness derivation order peak standard deviation standard deviation of a pure Gaussian profile standard deviation of a peak convolution with function T (49) (eq 25) exponential decay function (9)
LITERATURE CITED Kesner, L. F.; Giddings, J. C. High Performance Liquid Chromatography; Brown, P. R., Hartwich, R. A,, Eds.; John Wiley 8 Sons: New York, 1989; Chapter 15. Giddings, J. C. Anal. Chem. 1981, 53, 1170A. Giddings, J. C. Chem. Eng. News 1988, 66, 34. Caldwell, K. D. Anal. Chem. 1988, 60, 959A. Giddings, J. C.; Caldwell, K. D.; Jones, H. K. The Assessment of Particle Size Distribution; Provder, T., Ed.; ACS Symposium Series No. 332; American Chemical Society: Washington, D.C., 1987; Chapter 15. Kirkland, J. J.; Yau, W. W. Science 1982, 278. 121. Giddings, J. C. Sep. Sci. Technol. 1978, 73,3. Giddings, J. C. Dynamics of Chromatography. Pari I : Principles and Theory; Marcel Dekker: New York, 1965. Sternberg, J. C. A&. Chromatogr. 1966, 2 , 205. Grubner, 0. Anal. Chem. 1971, 43, 1934. Grubner. 0. Adv. Chromatogr. 1968, 6 , 173. Grushka, E. J. Phys. Chem. 1972, 76, 2586. Karaiskakis. G.; Myers, M. N.; Caldwell, K. D.; Giddings, J. C. Anal. Chem. 1981, 53, 1314. Giddings, J. C. Sep. Sci. Technol. 1984, 79, 831. Giddings, J. C.; Yang. F.-S. J. Colloid Interface Sci. 1984, 705. 55. Giddings, J. C.; Karaiskakis, G.; Caldwell, K. D.; Myers, M. N. J . Co//o&j Interface Sci. 1983, 92, 66. Kirkland. J. J.: Yau. W. W.: Doerner. W. A,: Grant. J. W. Anal. Chem. 1980, 52, 1944. Yang, F A . ; Caldwell, K. D.; Giddings, J. C. J. Collokj Interface Sci. 1983, 92,81. Giddings, J. C.; Yang, F. J. F.; Myers, M. N. Anal. Chem. 1974, 46, 1917. Jones, H. K.; Barman, B. N.; Giddings, J. C. J. Chromatogr. 1988, 455, 1.
Giddings, J. C.; Myers, M. N. Sep. Sci. Technol. 1978, 73,637. Caldwell, K. D.; Nguyen, T. T.; Myers, M. N. Sep. Sci. Techno/. 1979, 74, 935. Myers, M. N.; Giddings, J. C. Anal. Chem. 1982, 5 4 , 2284. Giddings, J. C.; Chen, X.; Wahlund, K.-G.: Myers, M. N. Anal. Chem. lg87, 59, 1957. Hansen, M. E.; Giddings, J. C.; Beckett, R. J. Colloid Interface Sci. 1989, 132, 300. Hansen, M. E.; Giddings, J. C. Anal. Chem. 1989, 61, 811. Mott, D. S.; Grushka, E. J. Chromatogr. 1978, 148, 305. Cram&, H. Mathematical Methods of Statistics, 7th ed.; Princeton University Press: Princeton, NJ, 1957; Chapter 17. Charlier. C. V. L. Applications de la Thgorie des Probabi//&s B / ' A s tronomie: Gauthier-Villars: Paris, 1931; Chapter 3. Edgeworth, F. Y. Proc. Cambridge Philos. SOC. 1905, 20, 36. Cram&, H. Skand. Aktuarietidskr. 1928, 7 1 , 13. Cram&. H. Random Variables and Probability Distributions; Cambridge University Press: Cambridge, U.K., 1962; Chapters 7 and 8. Gnedenko, 8. V.; Kolmogorov, A. N. Limit Distributions for Sums of Independent Random Variables; Addison-Wesley: Cambridge, MA, 1954 Chapter 8. Smirnov, V. Cows de Mathgmatiques Sup6rieurs; MIR: Moskow, 1972; Vol. 111, Chapter 6. Hildebrand, F. B. Introduction to Numerical Ana/ysis ; McGraw-Hill: New York, 1956; Chapter 7.
130
Anal. Chem. 1991, 63,130-133
(36) Dondi, F.; Betti, A.; 810, G.; Bighi. C. Anal. Chem. 1981, 53, 496. (37) Abramowitz, M.; Segun, I . Handbook of Mathematical function with formulas, eaphs and Mathematical Tables; Dover: New York, 1965. (38) Dondi, F. Anal. Chem. 1982, 54,473. (39) Dondi. F.; Remelli, M. J. Phys. Chem. 1988, 90, 1985. (40) Giddings, J. C. Gas Chromatography 7964:Goldup, A,, Ed.; Eisevier: Amsterdam, 1964; pp 3-25. (41) Giddings, J. C.; Eyring, H. J. Phys. Chem. 1955, 59,416. (42) Mc Quarrie, D. A. J. Chem. Phys. 1963, 38, 437. (43) Hovingh. M. E.: Thompson, G. H.; Giddings, J. C. Anal. Chem. 1970, 42, 195. (44) Giddings, J. C. J. Chem. Phys. 1968, 49,81. (45) Shure, M. R. Anal. Chem. 1988, 60,1109. (46) Dondi. F.; Puiidori, F. J. Chromatogr. 1984, 284, 293. (47) Betti, A.; Dondi, F.; Blo. G.; Coppi, G.: Cocco, G.; Bighi, C. J. Chromatogr. 1983, 259,433. (48) Dondi, F.; Remelli, M. J. Chromatogr. 1984, 315,67. (49) Remelli, M.; Blo. G.: Dondi, F.; Vidal-Madjar, M. C.; Guiochon, G. Anal. Chem. 1989, 67,1489. (50) 810,G.; Pedrlelli, F.; Remeili, M.; Balconi, L.; Sigon, F.; Dondi, F. Unpublished work. (51) Kirkland, J. J.; Yau, W . W.; Stoklosa, H. J.; Dilks, C. H., Jr. J. Chromatogr. Sci. 1977, 75,303. (52) Giddings. J. C. Anal. Chem. 1983, 35, 1999.
(53) Peterson, R. E., 11; Myers, M. N.; Giddings. J. C. Sep. Sci. Technol. 1984. 79,307. (54) Koch, T.; Giddings, J. C. Anal. Chem. 1986, 58, 994. (55) Lee, S.;Giddings, J. C. Anal. Chem. 1988, 60,2328. (56) Bracewell, R. N. The Fourier Transform and its Applications; McGrawHill: New York. 1986. (57) Maldacker, T. A.; Davis, J. E.; Rogers, L. B. Anal. Chem. 1974, 46, 637. (58) Programmer's Manual, 4th ed.; IBM Technical Publications Department: White Plains, NY, 1968; IBM Application Program, Systeml36O Scientific Subroutine Package (360A-CM-03X) Version 111, p 225. (59) Fletcher, R.: Reeves, C. M. Comput. J. 1964, 7(2), 149. (60) Littlewood, A. B. Gas Chromatography, 2nd. ed.; Academic Press: New York and London, 1970: p 158. (61) Giddings, J. C. Sep. Sci. Technol. 1978, 73,241. (62) Vidal-Madjar. C.; Guiochon, G. J. Chromatogr. 1977, 742, 61. (63) Chesler, S. N.; Cram, S. P. Anal. Chem. 1971, 43, 1922.
RECEIVED for review May 21,1990. Accepted October 8,1990. This work was supported by the Italian Ministry of the University and Scientific, Technological Research (MURST) and the Italian Research Council (CNR).
Isolation and Purification of Chlorophylls a and b for the Determination of Stable Carbon and Nitrogen Isotope Compositions Robert R. Bidigare,' Mahlon C. Kennicutt II,* and Wendy L. Keeney-Kennicutt Geochemical & Environmental Research Group, 833 G r a h a m Road, T e x a s A & M University, College S t a t i o n , T e x a s 77845 Stephen A. MackoZ D e p a r t m e n t of E a r t h Sciences, Memorial University, St. J o h n s , Newfoundland, Canada A l B 3 x 9
A method is described to Isolate and purify chlorophyll pigments from plant tissues for the determination of stable carbon and nltrogen isotope composltlons. Chlorophylls are initially isolated by solvent preclpltation, followed by preparative C18 reverse-phase high-performance liquid chromatography (HPLC) for final purlflcation. The purity of the isolated pigments (chlorophyll a and chlorophyll b ) was assessed by analytical HPLC, absorption spectroscopy, elemental analysis, and 'H NMR spectroscopy. A step-by-step evaluation of the separation indicates that molecular and isotopic integrity of the pigments are preserved during purification. Measurement of stable Isotope composition across the HPLC peak illustrated the necessity to collect the entire peak in order to maintain Isotopic integrity. Multiple stable Isotope compositions (613C, 615N) of Individual molecular markers (Le., pigments) are a powerful new tool to identify the sources of complex organic matter mixtures in the bio- and geospheres.
INTRODUCTION Organic matter in the bio- and geospheres is a complex mixture of living organisms and detrital remains. This complexity results from the multitude of source organisms, variable biosynthetic strategies available to organisms, and transfor-
* T o whom correspondence should be addressed. Present address: Department of Oceanography, 1000 Pope Rd, Universitv of Hawaii, Honolulu, HI 96822. *Present address: Department of Environmental Sciences, University of Virginia, Charlottesville, VA 22903.
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mations that occur during diagenesis and catagenesis. Chemical and stable isotopic characterization of organic matter can provide insight into its origin. However, the existing analytical techniques lack the specificity required to adequately characterize most natural mixtures. The stable isotope composition of bulk organic matter and molecular marker distributions are often difficult to interpret due to indistinct boundaries between sources and an inadequate understanding of source compositions (1-4). To resolve this complexity, the determination of the stable isotope composition of individual compounds has been investigated including, amino acids ( 5 ) ,photosynthetic pigments (6, 7 ) , monosaccharides (8), aliphatic and aromatic hydrocarbons (9-12), and geoporphyrins (10, 13-15). Chlorophyll-related compounds are ideal candidates for use in molecular stable isotope studies because (a) chlorophylls are the basis of the photosynthetic process that produces biomass, (b) tetrapyrroles contain two elements amenable to stable isotope analysis, (c) early diagenetic reactions are well defined, and (d) porphyrins and chlorins are abundant in nature. Tetrapyrrole-based diagenetic alteration products, geoporphyrins, are ubiquitous and stable in the geosphere as atested by their presence in coal and petroleum. The molecular and stable isotope compositions of individual geoporphyrins have been demonstrated to be useful in delineating precursor-product relationships for this class of compounds in geological samples (10, 14, 15). In contrast to the expanding data base on geoporphyrins, little or no information is available on the stable isotopic composition of pigments of extant plant species. This basic information is needed to model the formation and cycling of 0 199 1 American Chemical Society
