Anal. Chem. 1992, 64, 2459-2460
24bB
Outline of a Theory of Focusing in Linear Chromatography Leonid M.Blumberg Hewlett-Packard Company, Route 41, P.O.Box 900, Avondale, Pennsylvania 19311-0900
INTRODUCTION There is renewed interest in chromatographic focusing by simultaneous programming of column parameters in time and distance (along the column).'" Can such nonuniform (coordinate dependent) time-varying separation achieve better resolution than the separation in the equivalent uniform time-invariant separation? A theoretical solution to the problem was not found in the literature. Apparently, for any linear738 (independent of a solute concentration) separation, such improvementis not possible. The purpose of this short communication is to report the major results of the theory which led to that conclusion and to outline the flow of its logic. To stay close to the results of the previous work? this report, for the most part, is limited to the media where the distance-based velocitygradients are nearly constant within a zone. A more general study requires additional development of the theory for the evolution of a zone in a nonunifor medium, and the reexamining of such fundamental concepts as the resolution in a nonuniform separation, etc. These topics go beyond the capacity of a brief report and are being separatelyprepared for publication. Besides, a nonuniformity where the distance-based velocity gradients are nearly constant within the zones is practically the most important one. (More dramatic nonuniformity can significantly distort the zones or even split them.)
THEORY Throughout this paper, a chromatographic separation is assumed to be one-dimensional (all solutes follow the same path) and linear. It is important to emphasize that the properties of the medium where the separation takes place can be functions of distance and time. Consider a migration of two zones of different solutes. Let at a certain time, the distance, 1, between the zones be 1 = Zb - z. where za and Zb are the centers of masses of the zones, and the difference, Au, between velocities of the zones be Au = U b - u.. Assume that the solutes are almost identical, so that the zones remain just barely separated having coordinates, z. and Zb nearly equal to the coordinate z = (z, + Zb)/2 of the center of mass of both zones. Similarly,the difference of velocities of the zones is so small compared to the average u = (u, + Ub)/2that both u. and U b are nearly equal to u. Finally, both zones have nearly the same variances 62. The ratio9
their center of mass is at a coordinate z. The rate 1 dl 1 da --z(z-;z)
M E Re' = -
dz
of resolution tells how the separation of the zones evolves during their migration. Substitution of relation dlldz = (dl/ dt)(dt/dz) = Au/u and eq 1 into eq 2, and replacement of dddz by (da2/dz)/(2u) yields Au Re da2 R e ' = 4ua - - - - 2a2 dz
(3)
In further discussions, a sample introduction is said to be ideal if u2(0) = 0. A basic separation is a uniform timeinvariant one. An ideal basic separation is a basic separation with an ideal sample introduction. A separation is smooth if within the zones of solutes, the velocities of the solutes are nearly linear functions of distance for any z. In a basic separation, the velocity of a solute can be viewed as the sole representative of the identity of the solute. If Au, is the difference between the velocities of two solutes in a basic separation then the ratio 6 = Audu (the reduced difference in the velocities of the solutes) can be referred to as the identity gap between the solutes. Due to the possible nonuniformity of an arbitrary separation, additional difference in the velocities of the zones at a given time can come from the difference of the condition in the medium for the two solutes at that time. The latter can be represented by the local distance-based gradient, g = g(z), of velocity of the solutes at z. The net difference, Au, between the velocities of the two zones is the sum Au = Au, + gl = 6u + g l . Substitution of Au in eq 3 and taking account of eq 1yield (4)
Further simplificationof this expression is possible due to previous results.8 The model and derivations developed in ref 8 lead to the conclusion that for any smooth separation one has da21dz = H + 2a2g/u. In the rhs of this expression, the first term, H = H ( z ) ,is the plate height at z which reflects the dispersion in the medium while the second term reflects the change in a2 caused by the deformation of the zone due to the distance-basedvelocity gradients. Substitution of this expression into eq 4 results in the simpler relation
represents the distance-based resolution of the zones when (1)Rubey, W.A. J. High Resolut. Chromatogr. 1991, 14, 542-548. (2)Phillips, J. B.;Jain, V. Abstracts of papers presented at the 1991 Pittsburgh conference and exposition on analytical chemistry and applied spectroscopy;American ChemicalSociety: Washington,DC 1991; p 831. (3)Reighetti, P. G.; Gianzza,E.: Bianchi-Bosisio,A,;Sinha,P.; Kottgen, E. J. Chromatogr. 1991,569,197-228. (4)Van Puyvelde, F.;Chimovitz, E. H. J. Supercrit. Fluids 1990,3, 127-135. (5)Fuggerth, E.Anal. Chem. 1989,61,1478-1585. (6)Schomburg, G.; Roeder, W. J.High Resolut. Chrornatog. 1989,12, 218-225. (7)Guiochon,G.; Guillemin, C. L. Quantitative Gas Chromatography for Laboratory Analysis and on-line Process Control; Elsevier; Amsterdam, 1988. (8)Blumberg, L. M.; Berger, T. A. J. Chromatogr. 1992,596, 1-13. 0003-2700/92/0364-2459$03.00/0
The purpose of this study is to compare an arbitrary separation with the appropriately defined equivalent ideal basic separation. Notice that the local properties of a chromatographic medium are completely described by the two parameters8 H and u. Two media are z-equivalent to each other for a given zone of a solute if for that zone they have the same H ( z ) and u(z). The values R, = R,, ( z ) and R,' = RE,,'( z )are the resolution and its rate in the z-equivalent ideal basic separation. For such separation9 E = 62,u2 = Hz. (9)Giddings, J. C.Unified Separation Science: John Wiley & Sons: New York, 1991. 0 1992 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 20, OCTOBER 15, 1992
Substitution of these relations into eqs 1and 2, yields for R,, and R,,' 80
! I
40
1
I
!
I
I
...
I
These relations suggest that at any z, the specific resolution,
R,d6 (the resolution per unit of the identity gap), depends only on a(z) and H ( z ) while the specific rate of resolution, R.,'/8, depends only on With R,, and R,,', eq 5 can be rewritten as Ri/Rad = 2 - Rs/Rao, or
(7) Also, let E = E(z) = R$R,, -1 and B = t(z) = RiIR,; - 1 be excesses (relativeto the z-equivalentideal basic separation) of the resolution and its rate, respectively. Eq 7 becomes
E+e=O
(8)
RESULTS AND DISCUSSION
Equations 7 and 8 represent the main result of the theory. For simplicity in reasoning, the equations were derived for a medium with constant gradients of velocities of the solutes. It can be shown that these equations are valid for arbitrary velocities as well. In the general theory, however, the local values for plate height, the velocity, and its distance-based gradient should be replaced with their aggregate8 (effective) counterparts. Also the notion of resolution in a nonuniform separationand the entire strategy of comparisonof the quality of a nonuniform separation with that of a uniform separation needs to be more carefullyscrutinized. All these developments were beyond this brief report. Nevertheless,in the remaining, it is assumed that eqs 7 and 8 are valid for any linear separation. (The proof of the general case will be submitted for the publication later.) Equations 7 and 8 have important implications. They indicate that the sum of the reduced resolution, RJR,, and the reduced rate, RiIR,', of resolving any two solutes, as well as the sums of the excesses of these quantities are the important invariants of a separation. The sums always remain the same for any separation. Also, the ratios as well as the excesses E and t represent figures of merit for the local qualities of a separation and its evolution. According to eq 7,resolution of two solutes in any medium evolves in such a way that R$R, and RLIR,,' can never simultaneously exceed unity. If one is larger than unity, the other must be smaller than unity. Thus, at any location,the quality of the resolution and its rate can never simultaneously exceed similar qualities in the z-equivalent ideal basic separation. If one is better than the respective quality in the ideal basic separation, the other must be worse than its ideal basic counterpart. Similarly,the sum of the excesses of resolution and its rate is an invariant which always remains zero. If at any coordinate, z , resolution is excessive relative to the z-equivdent basic separation, its rate must have a deficiency (a negative excess). Conversely,only when the zones are underresolved (have a negative excess of resolution) can the rate of their resolution exceed that in the z-equivalent ideal basic separation. This discussion can be summarized in a form of the law of [an arbitrary] chromatographic separation (uniform or nonuniform, time-varying, or time-invariant).
Flgurs 1. Speclflc resolutlon, 4RJ6, vs coordinate, I (In meters), of the center of mass of two zones. All graphs represent evohrlng resolution of two solutes with the same identity gap, 6, In a 5-i-mlong column with H = 0.5 mm. The average, u, of velocltles of the zones Is also the same for all separatlons and does not change along the column. The dotted line, 4RJ8 = (I/H)I/~ represents the unlform timeInvariant separatlon with the ideal sample lntroductlon. The dashed lines, 44.18 = z/(u*(O)iHz)'I2,also reflects unlform tlme-lnvarlant separatlon but with the sample lntroductlon which took 10 times longer than the elutlon tlme of a nonretalned peak In the prevlous separation. The solld llne represents the separatlon with the same n o n h l sample Introduction as In the prevlous case, and with the focuslng by the negattve dlstance-based gradlent g = Cu where C = -1 m-I. The gradlent (nonunlformtty of the separatlon) travels along the column with the veloctty, u, of the zones. Hence,the zones always "see"the same dlstance-based velocity gradlent. However, at a ghren dlstance from the column Inlet, the dlstance-based gradlent changes with tlme providing for the nonunlform tlme-varylng separatlon.
Consider two one-dimensional linear separations of the same nearly identical solutes: an arbitrary separation A and a uniform time-invariant separation B with the ideal sample introduction. Let A and B be z-equivalent at a coordinute z. Then at z, the rate of resolving of the solutes in A cannot exceed the rate of resolving them in B unless the resolution in A is lower than the resolution in B.
Simply,no focusing can provide betterresolutionthan the one available from the equivalent ideal basic s e p aration. However, the negative implications of the theory should not be overestimated. While establishing the limits to the focusing, the theory helps to recognize its potential. After all, no separation is ideal and the room for the focusing always exists. A computer simulation of the focusing after a nonideal sample introduction is shown on Figure 1. Under the conditions of Figure 1, the nonideal sample introduction causes more than the 10-fold loss of the resolution compared to the ideal basic separation. The focusingprovidesa dramatic (better than 6-fold) recovery from the same nonideal sample introduction. ACKNOWLEDGMENT The author is indebted to Terry A. Berger and Raymond D. Dandeneau. Without Terry's assistance in this study and without Ray's support and encouragement,this work would not be possible. The author also appreciates stimulating discussion of chromatographic focusing with Georges Guiochon.
RECEIVED for review March 26, 1992. Accepted July 21, 1992.
