Anal. Chem. 1999, 71, 1645-1657
Limit of Detection, Dilution Factors, and Technique Compatibility in Multidimensional Separations Utilizing Chromatography, Capillary Electrophoresis, and Field-Flow Fractionation Mark R. Schure
Theoretical Separation Science Laboratory, Rohm and Haas Company, 727 Norristown Road, Spring House, Pennsylvania 19477
The theoretical development of the limit of detection (LOD) concept is extended from one separation column or channel used in gas, liquid, and size-exclusion chromatography, capillary electrophoresis, and field-flow fractionation to the case where multiple columns and channels are utilized in the so-called “comprehensive” and “heart cutting” modes of operation. Simple equations show that the column dilution factors and the split ratios used to compute the LOD are multiplicative per dimension. Thus, when more than one separation dimension is utilized in a sequential column mode of operation, a larger overall dilution of the original injected solute concentration will occur. The dilution of the initial zone in multidimensional geometrical separation systems such as planar two-dimensional thin-layer chromatography and two-dimensional slab gel electrophoresis is also considered. The practical aspects of this dilution are discussed with respect to the types of separation techniques that can be used to implement multidimensional separation systems. Multidimensional separation systems can be implemented in a number of ways which include planar or “geometric” systems and coupled-column systems. Examples of geometric systems include two-dimensional thin-layer chromatography (TLC),1,2 where successive one-dimensional TLC experiments are run at 90° angles with different solvent systems, and 2D electrophoresis, where gel electrophoresis is run in the first dimension followed by isoelectric focusing in the second dimension.3-5 Hybrids of these systems where chromatography and electrophoresis are used in each spatial dimension were reported nearly 40 years ago.6 (1) Consden, R.; Gordon, A. H.; Martin, A. J. P. Biochem. J. 1944, 38, 224232. (2) Grinberg, N.; Kala´sz, H.; Han, S. M.; Armstrong, D. W. In Modern ThinLayer Chromatography; Grinberg, N., Ed.; Marcel-Dekker Publishing: New York, 1990; Chapter 7. (3) O’Farrell, P. H. Biol. Chem. 1975, 250, 4007-4021. (4) Celis, J. E.; Bravo, R. Two-Dimensional Gel Electrophoresis of Proteins; Academic Press: New York, 1984. (5) Anderson, N. L.; Taylor, J.; Scandora, A. E.; Coulter, B. P.; Anderson, N. G. Clin. Chem. 1981, 27(11), 1807-1820. (6) Efron, M. L. Biochem. J. 1959, 72, 691-694. 10.1021/ac981128q CCC: $18.00 Published on Web 03/16/1999
© 1999 American Chemical Society
Coupled-column systems run in the so-called “comprehensive” method of separation accumulate an aliquot from a column or channel and inject this into the next separator in a sequential repeated real-time manner. Storage of the accumulating eluent is typically provided by sampling loops connected to an automated valve. Many variations on this theme exist which use various chromatographic and electrophoretic methods for one of the dimensions. In addition, the simpler “heart cutting” mode of operation takes the eluent from a first dimension peak or a few peaks and manually injects this into another column during the first dimension elution process. A partial compilation of these techniques is given in refs 7-15. Many of these combinations have been tabulated and discussed previously16 in terms of compatibility and practicality. These multidimensional separation systems have also been categorized16 as simultaneous displacement, which may be implemented with planar separation systems, and sequential displacement systems, which may be implemented with coupled columns. We will not use this terminology here because it does not uniquely delineate the two types of systems. Both geometric and coupled-column configurations can be separating simultaneously in multiple dimensions. Additionally, both modes can also be run one dimension at a time. In these multiple separation systems two effects are primarily exploited to obtain more desirable separations. These are an increase in selectivity and an increase in peak capacity. The (7) Balke, S. T. Quantitative Column Liquid Chromatography Journal of Chromatography Library; Elsevier Publishing: New York, 1984; Vol. 29. (8) Bushey, M. M.; Jorgenson, J. W. Anal. Chem. 1990, 62, 161-167. (9) Cortes, H. J. Multidimensional Chromatography: Techniques and Applications; Marcel Dekker: New York, 1990. (10) Larmann, J. P.; Lemmo, A. V.; Moore, A. W.; Jorgenson, J. W. Electrophoresis 1993, 14, 439-447. (11) Liu, Z.; Sirimanne, S. R.; Patterson, D. G.; Needham, L. L.; Phillips, J. B. Anal. Chem. 1993, 66, 3086-3092. (12) Kilz, P.; Kru ¨ ger, R.-P.; Much, H.; Schulz, G. In Chromatographic Characterization of Polymers: Hyphenated and Multidimensional Techniques; Provder, T., Urban, M. W., Barth, H. G., Eds.; Adv. Chem. Ser. No. 247; American Chemical Society: Washington, D.C., 1995. (13) Venema, E.; de Leeuw, P.; Kraak, J. C.; Poppe, H.; Tijssen, R. J. Chrom. A 1997, 765, 135-144. (14) Murphy, R. E.; Schure, M. R.; Foley, J. P Anal. Chem. 1998, 70, 15851594. (15) Murphy, R. E.; Schure, M. R.; Foley, J. P. Anal. Chem. 1998, 70, 43534360. (16) Giddings, J. C. Anal. Chem. 1984, 56, 1258A-1270A.
Analytical Chemistry, Vol. 71, No. 8, April 15, 1999 1645
increase in selectivity is especially beneficial when a specific separation mechanism can be invoked to further resolve some part of the analyte structure, for example, the separation and resolution of mixed alkyl and ethylene oxide (EO) block copolymers into the individual alkyl and EO distributions by normal phase and reversed phase liquid chromatography (NPLC and RPLC).15 The increase in peak capacity is useful when specific resolution of the analyte molecular structure is overwhelmingly difficult or when the sample is too complex for specific structure retention. One example of this case is the separation of tryptic digest components by RPLC in the first dimension followed by capillary electrophoresis (CE) in the second dimension.10 Another example of this is the use of multiple column gas chromatography (GC) for the analysis of pesticides.11 The peak capacity for two coupled columns is approximately the product of each individual technique,17,18 and this coupling may offer some relief in resolving the large degree of peak fusion found in single dimension separation techniques.19 These multidimensional comprehensive mode separations appear to be a very forward-looking step in the evolution of analytical separation science. As mentioned above, many of these multidimensional techniques have been implemented in the planar or geometric mode many decades ago. The main problem with using the planar methods, despite the sophistication in coupling two complementary separation techniques, is the difficulty in detection and collection of zones among other less critical problems, such as slab preparation. The detection problem, although easier for column methods because the detection region is fixed in space, is not without other problems for the coupledcolumn techniques. These include the slowing of the first dimension separation system for comprehensive systems so that first dimension peaks may be sampled an adequate number of times by the next dimension separation system. This aspect has been recently studied in detail.14 Another problem with coupledcolumn systems is one of zone dilution by each system comprising a separation dimension. We have seen this as a problem that was insurmountable when coupling electrical field-flow fractionation (electrical FFF) and LC methods20 using various sensitive detection schemes such as the evaporative light scattering detector. However, this appears to be a general problem that warranted further study as no theoretical treatment appears to have been produced that relates the detection limits and column-induced zone dilution factors to the various multidimensional separation systems based on comprehensive, heart cutting, and geometric technologies. In this paper the theory to the limit of detection and column dilution for single dimension geometric separation systems and single columns is extended to include multidimensional separation systems. Chromatographic, electrophoretic, and FFF techniques are included in the coupled-column study where the dilution of zones at the elution peak center is studied. It will be shown for (17) Guiochon, G.; Gonnord, M. F.; Zakaria, M.; Beaver, L. A.; Siouffi, A. M. Chromatographia 1983, 17, 121-124. (18) Guiochon, G.; Beaver, L. A.; Gonnord, M. F.; Siouffi, A. M.; Zakaria, M. J. Chromatogr. 1983, 255, 415-437. (19) Davis, J. M. In Advances in Chromatography; Grushka, E., Brown, P. R., Eds.; Marcel Dekker Publishing: New York, 1994; Vol. 34. (20) Palkar, S. A.; Murphy, R. E.; Schure, M. R. In Particle Size Distribution III: Assessment and Characterization; Provder, T., Ed.; Symp. Ser. No. 693; American Chemical Society: Washington, D.C., 1998.
1646 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
both multidimensional geometric and coupled-column systems that the dilution factors are multiplicative resulting in severe detection problems when certain techniques such as FFF are used as one of the dimensions in a multidimensional scheme. In addition, for coupled column systems the column split ratio is also multiplicative, which causes further zone dilution. This places constraints on the types of techniques which may be coupled to form a practical multidimensional separation system. THEORY One-Dimensional Column Limit of Detection. The theory of the limit of detection has been outlined in a number of papers pertaining to chromatography21-23 and chemical analysis.24-26 The essential components are as follows. Note that we will only consider Gaussian peaks in this paper. The concentration at the peak maximum, Cmax, as determined by a concentration-specific detector, is given by
Cmax) minj/Vdil
(1)
where minj is the mass of solute injected and Vdil is the dilution volume. The dilution volume is the volume of fluid that the zone has occupied due to being subjected to various zone broadening processes. Since different zone broadening mechanisms are present in the different techniques that are discussed in this paper, the dilution volume has an operational definition given by rearranging eq 1 to the simple form Vdil ) minj/Cmax. The minimum detectable concentration at the peak maximum, q Cmax , is given by
Cqmax ) 5(An/Sd) ) 5Φ
(2)
where An is the noise amplitude expressed as either a peak-topeak or root-mean-square (rms) amplitude21-26 and Sd is the detector sensitivity. The detector sensitivity is given in units of amplitude per unit concentration, much as a generalized sensitivity is given as the change in output per unit input. The factor of 5 in eq 2 is arbitrary and subject to debate.21-26 Combining eqs 1 and 2 we can define a limit of detection (LOD) as a mass amount, mqinj, for which
mqinj ) 5‚Φ‚Vdil
(3)
In eq 3, Φ is detector specific while Vdil is dependent on the specific separation technique. Thus, we will focus on the Vdil term as the separation-based term of importance. Dilution in Chromatography. The equation of a Gaussian peak with the solvent volume as the independent variable and concentration as the dependent variable is a convenient starting point for the derivation that follows. At the peak maximum, where (21) Karger, B. L.; Martin, M.; Guiochon, G. Anal. Chem. 1974, 46, 1640-1647. (22) Foley, J. P.; Dorsey, J. G. Chromatographia 1984, 18, 503-511. (23) Pal., F.; Pungor, E.; sz Kovats, E. Anal. Chem. 1988, 60, 2254-2258. (24) Hubaux, A.; Vos, G. Anal. Chem. 1970, 42, 849-855. (25) Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 55, 712A-724A. (26) Taraszewski, W. J.; Haworth, D. T.; Pollard, B. D. Anal. Chim. Acta 1984, 157, 73-82.
the volume is equal to the retention volume of the peak, i.e., V ) VR:23
Cmax )
[
]
minj minj (V - VR)2 minj ) exp ) (4) 2 Vdil x2πσ 2σv x2πσv v
where σv is the Gaussian standard deviation in volume units. To further develop this problem we can utilize relationships for the number of theoretical plates N such that
N ) (VR/σv)2 ) (t′/σ)2
(5)
where t′ is the mean elution time and σ is the Gaussian standard deviation in time units. Note that VR ) F‚t′ and σv ) F‚σ where F is the flow rate. Combining eqs 4 and 5 yields
Vdil ) VRx2π/xN
Vdil ) wVRx2πχ〈v〉/LD
Vdil ) 2πbw2xηd〈v〉Lλ/kBT
where Cinj is the solute concentration injected with volume Vinj. Furthermore, the ratio Vdil/Vinj is the dilution factor f. It is easily seen from eqs 3 and 7 that the larger the dilution volume and the dilution factor, the larger will be the mass limit of detection. With large dilution factors and dilution volumes the concentration at the peak maximum, Cmax, will be smaller. Hence, the lowest dilution volume and dilution factor is most desirable in making mqinj as small as possible. The limit of detection may be minimized through two pathways. First, the detector specific Φ factor should be as small as possible by utilizing a low noise and high sensitivity detector. Second, as embodied in eq 6, the separation system should have a high plate count and be run at the lowest possible retention volume. Both of these aspects are important and become critical in practical applications of coupling multidimensional techniques, as will be discussed below. Dilution in FFF. The functional form of eq 6 is suitable for the study of zone dilution with elution processes such as chromatography and FFF because the retention volume has a well understood physical meaning. Using eq 6, Vdil has been previously derived for FFF27 for a number of cases. For the general case, the particle diameter, d, for colloidal particles and the molecular weight, Mw, for polymers are related to the diffusion coefficient (27) Schure, M. R.; Weeratunga, S. K. Anal. Chem. 1991, 63, 2614-2626.
(9)
In eq 9, b is the channel breadth, η is the viscosity, λ is the nondimensional mean layer thickness,29,30 kB is Boltzmann’s constant, and T is the absolute temperature. Dilution in CE. There appears to be no dilution volume equation for CE in the literature so one will be derived here. We can utilize eq 4 to get a form suitable for CE by noting that σv ) A‚σL where A is the capillary cross-sectional area and σL is the Gaussian standard deviation in length units. Hence eq 4 can now be written as
Cmax )
(7)
(8)
where w and L are the channel width and length, and 〈v〉 and χ are the average fluid velocity and the nondimensional nonequilibrium coefficient.28-30 For the specific case of a well-retained colloidal particle:27
(6)
which has been previously presented for the dilution volume.21-23 If it is assumed that the solute mass minj is conserved throughout the separation process, then:
Cmax Vinj 1 ) ) Cinj Vdil f
D and the retention volume VR so that:
minj Ax2πσL
(10)
Combining the definition of the plate height H in a uniform column as σ2L/L and rearranging, noting that H ) L/N, gives σL ) L/xN. Note that L here is the capillary length from injector to detection window. Substitution of σL ) L/xN and A ) πdc2/4 (dc is the column inner diameter) into eq 10 and solving for Vdil through the relation Vdil ) minj/Cmax (eq 4) yields
Vdil )
πx2πdc2L 4xN
≈
2dc2L
xN
(11)
It is noted that the retention volume in eq 6 is equal to the capillary volume in eq 11. This equality is expected because in the case discussed here for ideal CE where adsorption effects are absent there is no retention and broadening occurs strictly over one capillary volume of solute migration. The result of eq 11 will be used below to compute Vdil in both single and multiple dimension separation systems which incorporate CE in one of the dimensions. In all of the treatments of the various techniques discussed in this paper we include the finite injection volume as a component that affects the technique efficiency. This aspect has been discussed previously for chromatography.21 Although this effect will usually modify N by just a few percent for chromatographic and FFF techniques, the injection volume is an extremely important consideration in judging the efficiency of CE methods.31 Hence, the conditions for CE will include the injection volume as an important component of the efficiency and in the subsequent (28) Giddings, J. C.; Yoon, Y. H.; Caldwell, K. D.; Myers, M. N.; Hovingh, M. E. Sep. Sci. 1975, 10, 447-460. (29) Caldwell, K. D. Anal. Chem. 1988, 60, 959A-971A. (30) Martin, M. In Advances in Chromatography; Marcel Dekker Publishing: New York, 1998; Vol. 39, Chapter 1. (31) Huang, X.; Coleman, W. F.; Zare, R. N. J. Chromatogr. 1989, 480, 95-110.
Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
1647
Figure 1. A schematic diagram of the components and terminology used to describe multiple column systems for multidimensional separations.
calculation of f in CE. The general treatment for including finite injection volume in the calculation of N is developed below. Coupled Columns. For the case of M coupled columns used in the comprehensive mode or heart cutting mode of operation, the eluent from column i is typically split and the remaining eluent stored in sample loop i + 1. When the sample loop is filled, the eluent is injected into column i + 1. The eluent from the last column in this scheme typically enters the detector with no further splitters or sample loops. This is shown schematically in Figure 1 along with the subscript notation used in this treatment. The coupled-column separation systems described to date typically produce more column effluent than can be injected at any one time without splitting. To reduce the eluent volume, a split-flow system of some sort or a solute concentrator is necessary to match the injection volume requirements of the next column. Concentrating the solute through on-column concentration will be discussed below. Solute mass is lost in the schemes which employ some form of splitter. We account for this loss by including a term denoted as Si, which is defined as
Si ) Vi-1,s/Vi,inj
Si ) 8σv/N′Vi,inj
(13)
Equation 5 can be rearranged to yield σv ) VR/xN, which when substituted into eq 13 gives
Si ) 8VR/N′xNVi,inj
(14)
which is useful for all of the techniques discussed in this paper noting that for CE the retention volume is simply the capillary volume, as discussed above. Noting that minj ) Cinj‚Vinj, eq 4 gives
C1,max )
C0,inj‚V0,inj C0,inj ) S1‚V1,dil S1 f1
(15)
(12)
where Vi-1,s is the volume from process i - 1 per sampling period, which will be split to give a volume of eluent Vi,inj to process i per sampling period. The subscript refers to the dimension or column number that receives solute from this split, as shown in Figure 1. The subscript 0 refers to the initial injection mass, volume, and concentration. The volume of eluent across a peak is assumed to be equal to 8σv, and this volume will be sampled N′ times across the peak. This is similar to the treatment given in ref 14; however, we use the symbol N′ to distinguish the number of samples from the number of theoretical plates N. Hence the volume of eluent per sampling period from column i - 1, Vi-1,s, is equal to 8σv/N′. This volume must be split to accommodate a sampling loop 1648 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
volume of Vi,inj. The split ratio for matching a column to a sampling valve is therefore
where fi ) Vi,dil/Vi,inj as is consistent with eq 7. This treatment can be extended to two columns (and subsequently M columns) if we assume that the maximum injection concentration for column two is just C1,max. Now the injection volume into the second column is determined by the sample loop volume in valve 2 and
C2,max )
C1,max‚V1,s C1,max C0,inj ) ) S2‚V2,dil S2 f2 S1 f1‚S2 f2
(16)
The product of split ratios and column dilution factors, as shown in the denominator of the right-hand side of eq 16, will be referred to as the net dilution factor.
Valve Sampling. An assumption inherent in eq 16 is that each successive dimension is sampled frequently enough that the concentration of solute in the sample loop is equal at some point to the Cmax of the accumulating solute peak. This is an approximation, and a simple treatment using the mathematics described in ref 14 can be applied to achieve more detail and rigor. Toward this end we further assume that sampling will be symmetrical around the peak maximum. This will never be rigorously true but provides a convenient starting point for this simple treatment. As stated previously, Gaussian peaks are assumed here; however, peaks with t′ ) 0 are used in this treatment as a mathematical simplification so that
C(t) )
1
e-t /2σ 2
2
(17)
x2πσ
The average value, hf, of a time-varying continuous function f(t) can be obtained between -tq and tq as
hf )
∫
1 2tq
tq
-tq
f(t) dt
(18)
The Gaussian profile given in eq 17 can be substituted into eq 18 so that f(t) ) C(t). The average of the function C(t) can be determined between -tq and tq by using the well-known relationship32
∫e x2πσ 1
u -t2/2σ2
0
[ ]
u 1 dt ) erf 2 x2σ
Table 1. Relative Average Concentration Θ as a Function of the Number of Samples N′ from Eq 21 N′
Θ
N′
Θ
1 3 5
0.313 0.769 0.903
7 9 11
0.948 0.968 0.978
Some tabulated values of Θ as a function of N′ are given in Table 1. As can be seen from Table 1, for N′ g 3, there is only about a 25% or less reduction in Cmax due to the averaging process in the loop. Symmetrical sampling represents a best case scenario; for nonsymmetrical sampling near the peak top, Θ will be slightly smaller than the values given in Table 1. It was recently determined14 that one should sample at least four samples across a peak to get reasonably high resolution and to avoid “phase error” whereby the zone shape strongly depends on where sampling starts. Hence, the concentration drop near the peak maximum due to loop sampling is a small perturbation on the ideal Cmax value. The Θ factor given in eqs 20 and 21 can be incorporated into eq 16 to include the averaging from sample collection. We now write eq 16 in recursive form and include Θi as the fractional reduction in Ci,max due to sampling:
Ci,max )
Ci-1,max Vi,inj Ci-1,max‚Θi ‚Θi‚ ) Si Vi,dil Si‚fi
(19) This can be generalized to an M dimensional separation system: M
where erf is the error function32 and u is a general variable. The one-sided integral given in eq 19 can be doubled to give the symmetric average concentration for zones collected between -tq and tq. Hence, the average concentration relative to the concentration maximum is obtained by using symmetry and eqs 17 through 19 so that
[ ]
[ ]
x2πσ x2π C h tq Γ )Θ) erf ) erf q Cmax 2Γ 2t x2σ x2
(20)
where Γ is the nondimensional ratio tq/σ and C h is the average concentration. This problem can be further reduced by relating N′, defined previously as the number of samples injected into the next dimension separation system across a peak width of 8σ, to the sampling time, ts, so that ts ) 8σ/N′. The choice of 8σ as a measure of the peak baseline has been discussed previously.14 To preserve symmetry we consider that N′ ) 1, 3, 5, ... so that the sample is always centered on the peak and under this condition 2tq ) ts and Γ ) 4/N′ so that
Θ)
[ ]
x2πN′ 2x2 erf 8 N′
(22)
(21)
(32) Handbook of Mathematics; Chemical Rubber Company: Boca Raton, FL, 1997.
Θi
∏S ‚f
CM,max ) C0,inj
i)1
(23)
i i
Since the last column or channel typically feeds the detector without a splitter and sampling valve, CM,max represents the concentration that the detector will sample. Equation 23 shows the general result that the dilution factors, fi, and split ratios, Si, are multiplicative for an M-dimensional separation system. Hence when coupled-column techniques are utilized it is imperative that small dilution factors and small split ratios be present yielding a small net dilution factor. This places limits on both the types of separation systems that can be coupled and the type of detectors that can be employed. Detectors such as the laser-induced fluorescence (LIF) detector, the mass spectrometer (MS), and the flame ionization detector (FID) used in GC are particularly sensitive and useful for multiple columns. For polymer analysis the evaporative light scattering (ELS) detector is also very sensitive and provides an easy to use capability. However, the ELS detector is not particularly sensitive as a small molecule detection system and this becomes an issue when multidimensional separations are utilized. Detectors such as the refractive index (RI) detector for LC and the thermal conductivity detector for GC may not be suitable in this configuration. The common UV detector, which is sensitive for aromatics and some other chromophores, may lack the sensitivity for general usage when coupled columns are utilized. Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
1649
Coupled-Column Limit of Detection. The compact form of eq 23 can further be developed into an explicit form for the mass LOD, mqinj, which is now written for multidimensional systems as q m0,inj and
Cq0,inj ) mq0,inj/V0,inj
(24)
q where C0,inj is the initial injection concentration at the detection limit. By substitution of eq 24 into eq 23, the peak maximum q concentration Cm,max at the detection limit is defined for the Mth column as
q CM,max )
mq0,inj V0,inj
Θi
M
∏ S ‚f i)1
(25)
i i
q Solving for m0,inj gives
q mq0,inj ) CM,max ‚V0,inj
M
Si‚fi
i)1
Θi
∏
(26)
These equations can be compressed to give the form of the one-dimensional counterpart LOD given in eq 3 by the following q considerations. The definition of Cmax given in eq 2 is dimenq sionally invariant, i.e., CM,max ) 5Φ no matter how many columns are involved, as this equation is a statement of how the concentration limit of detection is to be defined. This then gives a general statement of the mass detection limit for multidimensional systems as M
mq0,inj ) 5‚Φ‚V0,inj
∏
Si‚fi
i)1
Θi
(27)
Defining the net dilution volume, VTdil, as
VTdil ) V0,inj
M
Si‚fi
i)1
Θi
∏
(28)
allows the direct production of the multidimensional analogue of eq 3 now written as
mq0,inj ) 5‚Φ‚VTdil
(29)
Equations 23 and 27 clearly identify the two effects that lead to the overall loss of solute concentration and the elevation of the detection limit. These effects are the dilution volume, which is an intracolumn effect, and the split ratio, which is an intercolumn effect. Since the net dilution factor is the product of these intracolumn and intercolumn effects, the net dilution factor is most important. If either the column dilution factors or the split ratios are large, the net dilution factor can make the coupling of columns impractical. If both are large, even the most sensitive detector will not be able to form a practical separation system. In some cases one of the separation systems may be able to enrich the solute at the head of a column or channel by keeping 1650 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
solvent (LC), temperature (GC), field (FFF), or some other operational variable at some temporary value prior to elution. In these cases the injection volume Vi,inj can be much larger than is typically utilized and the split ratio Si much smaller. This will lead to a reduction in the product Si‚fi and will facilitate an increase in q as denoted by eqs 23 and 27, CM,max and a decrease in m0,inj respectively. Some separation systems are better at allowing this enhancement, for example, multidimensional GC where the sampling valve and second column inlet can be integrated11 to allow on-column enhancement with no splitting of the sample volume. On-Column Sample Volumes. Finite injection volume is an independent additional contribution to zone broadening and its extent on the calculations used in this paper must be quantified, especially in the case of CE as will be discussed shortly. The effect of injection volume on the LOD has been reported previously.21 It was shown that larger injection volumes can lower the LOD at the expense of broader zones. We will show the extent of the zone broadening increase due to finite injection volume through the fractional reduction in the number of available theoretical plates. The equations used for this evaluation are as follows. For a plug injection it is well-known that the length-based zone variance contribution from injection is given as33
σinj2 ) Linj2/12
(30)
where Linj is the length of the injected plug within the column and can be expressed as
Linj ) LVinj/Ve
(31)
In eq 31 Ve is approximately the sum of the interstitial and pore volume and the volume of solute in the stationary phase. However, in chromatography the initial segment of the column is typically overloaded. Hence, the solute will not be retained at the nearequilibrium value as in the downstream regions. For this reason and as previously discussed,21 it is most difficult to assign an exact value to Ve. Since the volume consideration as used here is not critical to the accuracy of the results nor the central meaning of the paper, we will simply assume for this treatment that Ve is the physical volume available to the solute, i.e., the column volume multiplied by where is the fraction of total column volume contained external to the surface of the particle. The exception to this is GPC where we will use the retention volume for Ve. Since the total zone variance is comprised of variances from individual sources, we note that σT2 ) σinj2 + σL2, where σT2 is the total zone variance and σL2 is due to purely intracolumn separation processes. The total number of theoretical plates is thus
NT )
L2 σinj2 + σL2
(32)
(33) Sternberg, J. C. In Advances in Chromatography; Marcel Dekker Publishing: New York, 1966; Vol. 2, Chapter 6.
Utilizing Ninj ) L2/σinj2 and Nsep ) L2/σL2 it can easily be seen that
NT )
1 1 1 + Ninj Nsep
(33)
Table 2. Summary of Conditions for the Calculation of Dilution Factors for LC and GC technique LC
The loss of total plates due to injection processes can be expressed as a fraction, φ, so that φ ) 1 - (NT/Nsep). This gives a convenient metric to describe the loss in available plates when finite volume injections are to be considered. By using the equations given above it is easy to show that
φ)
1 1 ) Ninj σL2 1+ 1 + Nsep σinj2
(34)
For the calculations given in the paper we use NT wherever the number of theoretical plates is called for. In this way the effect of finite volume considerations is accounted for. This will be seen to be a very minor effect for most methods but extremely important for CE. RESULTS AND DISCUSSION Technique-Dependent Dilution Factors. In this section we use the previously developed mathematics and compute some typical column dilution factors for coupling a variety of techniques. First, we compare the single column results to get an estimate of how the dilution varies among the three techniques of chromatography, CE, and FFF. The numbers used for these calculations are subjective, and in this regard we offer these as reasonable estimates based on previous experience, column manufacturer’s literature, and previous literature values. References are given where specific values were used from specific multidimensional systems. The retention volume in chromatography is easily calculated by the ratio V0/R, where V0 is the solvent accessible column volume and R is the well-known retention ratio equal to t0/t′ where t0 is the void time. The capacity factor k′ is related to R via R ) 1/(1 + k′). For the case of a packed column used typically in LC
VR ) V0‚(1 + k′) ) πdc2(1 + k′)L/4
(35)
Note that ) b + (1 - b)‚p, where b is the fraction of column volume contained in the interstitial region and p is the fraction of particle volume contained in the particle pore. Typical values of b are 0.4,34 and typical values of p range between 0.3 and 0.7.34 Hence, for parameters typical of a porous silica packing material ) 0.8. For parameters typical of LC where the column diameter dc ) 4.6 mm, ) 0.8, L ) 25 cm, and k′ ) 2, then VR ) 9.97 mL. Furthermore, if Nsep ) 10000 and Vinj ) 10 µL, the dilution factor f is equal to 25.1. This represents a good column for LC. Previously,21 a dilution factor of approximately 100 was suggested for LC. The parameters used for this calculation are summarized in Table 2 under the column labeled first dimension. The (34) Guiochon, G.; Shirazi, S. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: New York, 1994.
GC
first dimension
second dimension
dc ) 4.6 mm dp ) 5 µm k′ ) 2 L ) 25 cm Nsep ) 10000 Vinj ) 10 µL ) 0.8
dc ) 4.6 mm dp ) 3 µm k′ ) 2 L ) 3 cm Nsep ) 1500 Vinj ) 15 µL ) 0.8
f ) 25.1 Vdil ) 0.25 mL VR ) 9.97 mL φ ) 0.00749
f ) 5.60 Vdil ) 0.0840 mL VR ) 1.20 mL φ ) 0.150
dc ) 0.25 mm k′ ) 2 L ) 25 m Nsep ) 125000 Vinj ) 1 µL
dc ) 0.1 mm k′ ) 2 L)1m Nsep ) 5000 Vinj ) 0.1 µL
f ) 26.2 Vdil ) 0.0262 mL VR ) 3.68 mL φ ) 0.00687
f ) 8.63 Vdil ) 0.863 µL VR ) 23.6 µL φ ) 0.0633
parameters used for the second dimension column calculation given in Tables II-IV and VI will be discussed below in the section entitled Combinations of Techniques. Wherever possible three significant digits of precision are used throughout the calculations reported in the tables and in this paper. If a smaller number of significant digits is used it is implied that there are at least three places of significance to the number. In these tables the input parameters to the equations are given first in the columns followed by the calculated parameters which are typically f, Vdil, VR, and φ. The tables are computed by using a series of small FORTRAN programs. Two other types of chromatography, namely size-exclusion chromatography (SEC) and GC, have been utilized in the multidimensional format. We can calculate some typical dilution factors for these techniques in a manner similar to the LC calculation given above. For GC, we calculate the dilution factor with a wall-coated capillary column using eq 35 where dc is much greater than the film thickness so that ) 1. For the conditions given in the first column of Table 2 for GC, which are typical of capillary GC, we find that the dilution factor f is nearly the same (f ) 26.2) as that found for the LC case. This is interesting because the parameters are so different. The calculation of the dilution factor for SEC is based on parameters found in the SEC literature. These are listed in the first column of Table 3. The retention volume in SEC is calculated as VR ) V0 + KSEC‚Vp, where V0 is the interstitial pore volume external to the particle pore and Vp is the internal pore volume accessible to the solute. Typical values of V0 and Vp are 5.2 and 6.0 mL, respectively. The equilibrium constant KSEC varies from 0 where the solute is fully excluded from the pores to 1 where the solute is fully accessible to the pores. For the case KSEC ) 0.5, VR ) 8.20 mL. By using these numbers and the other numbers from the first column in Table 3, the dilution factor f is calculated to be 6.54. This number is smaller than for the other two chromatographic techniques because the injection volume is much larger than for the other techniques. This is because SEC has a large stationary-phase Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
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Table 3. Summary of Conditions for the Calculation of Dilution Factors for the SEC and FFF Techniques technique SEC
FFF
first dimension
second dimension
dc ) 8.0 mm dp ) 5 µm Ksec ) 0.5 L ) 30 cm Nsep ) 1000 V0 ) 5.2 mL Vp ) 6.0 mL Vinj ) 100 µL
dc ) 8.0 mm dp ) 3 µm Ksec ) 0.5 L ) 5 cm Nsep ) 250 V0 ) 0.9 mL Vp ) 1.0 mL Vinj ) 50 µL
f ) 6.54 Vdil ) 0.654 mL VR ) 8.20 mL φ ) 0.0122
f ) 4.50 Vdil ) 0.225 mL VR ) 1.40 mL φ ) 0.0259
b ) 2.0 cm d ) 0.2 µm F ) 1 mL min-1 L ) 60 cm Vinj ) 2 µL w ) 127 µm λ ) 0.0095
b ) 1.0 cm d ) 0.2 µm F ) 0.25 mL min-1 L ) 60 cm Vinj ) 2 µL w ) 62.5 µm λ ) 0.0095
f ) 1340 Nsep ) 651 Vdil ) 2.68 mL VR ) 27.3 mL φ ) 9.34 × 10-5
f ) 164 Nsep ) 2640 Vdil ) 0.327 mL VR ) 6.71 mL φ ) 0.00623
volume compared to GC and LC. Hence, the larger sample capacity gives a smaller dilution factor. By using eq 9, the dilution factor for FFF can be evaluated. We use parameters typical of a colloidal particle separation used in electrical FFF, flow FFF, or a sedimentation FFF fractionator, where L ) 60 cm, d ) 0.2 µm, F ) 1 mL min-1 (〈v〉 ) 0.66 cm s-1, λ ) 0.0095 (R ) 0.056), b ) 2.0 cm, w ) 127 µm, and Vinj ) 2 µL. With these parameters the dilution factor f is equal to approximately 1340, as summarized in Table 3. This is quite large but expected. This large figure could be substantially reduced if the channel width could be reduced by one-half. This is technologically difficult but not impossible. Under the most ideal conditions where the channel breadth, b, is reduced to 1 cm, the channel width w is 62.5 µm, and a flow rate of 0.25 mL min-1 (〈v〉 ) 0.66 cm s-1) is utilized, keeping the other parameters the same as above gives a dilution factor of approximately 164. Under these conditions FFF gives about eight times the dilution expected from LC. Channels of these width values are exceedingly rare due to the difficulty in construction. Another way to decrease the dilution is to use a dual outlet where the wall-resident flow is separately removed from the main channel solvent, which is mostly absent of solute.35,36 Although this may reduce the dilution factor by a factor of approximately 6, these outlets are difficult to maintain, are rarely utilized in most laboratories where FFF equipment is found, and have not been offered in commercial FFF equipment yet. However, this technique and others when applied in the future may bring the dilution factor for FFF more in range of the dilution factors found in chromatographic equipment. For CE the dilution volume is much lower than FFF and chromatographic techniques. Using eqs 7 and 11 and the typical (35) Wahlund, K.-G.; Winegarner, H. S.; Caldwell, K. D.; Giddings, J. C. Anal. Chem. 1986, 58, 573-578. (36) Li, P.; Hansen, M. E.; Giddings, J. C. J. Microcolumn Sep. 1998, 10(1), 7-18.
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Table 4. Summary of Conditions for the Calculation of Dilution Factors for the CE Technique technique CE with short Linj
CE with medium Linj
first dimension
second dimension
L ) 40 cm Linj ) 0.25 mm Nsep ) 100000 Vinj ) 0.491 nL dc ) 50 µm t′ ) 5 min
L ) 10 cm Linj ) 0.25 mm Nsep ) 25000 Vinj ) 0.491 nL dc ) 50 µm t′ ) 1.25 min
f ) 12.7 Vdil ) 6.24 nL φ ) 0.00324
f ) 6.38 Vdil ) 3.13 nL φ ) 0.0128
L ) 40 cm Linj ) 1.00 mm Nsep ) 100000 Vinj ) 1.96 nL dc ) 50 µm t′ ) 5 min
L ) 10 cm Linj ) 1.00 mm Nsep ) 25000 Vinj ) 1.96 nL dc ) 50 µm t′ ) 1.25 min
f ) 3.25 Vdil ) 6.39 nL φ ) 0.0495
f ) 1.74 Vdil ) 3.42 nL φ ) 0.172
values of dc ) 50 µm, L ) 40 cm, Nsep ) 100 000, and Vinj ) 0.491 nL, the dilution factor f is approximately 13. This is summarized in Table 4. Note that we provide two calculation sets here, one with a short Linj and the other with a factor of 4 larger Linj. As can be seen from the φ values, the choice of Linj is quite important as the net separation efficiency is critically tied to Linj. This is most important in the second dimension numbers for CE shown in Table 4 where a loss of 17.2% in plates is shown for the smaller length and medium injection length column. The dilution factor is very low for CE when a loss in plates is taken because the injection volume is increased. This appears to be a compromise that must be taken to effectively utilize CE in even the single column format. Overall, the dilution factor of CE is approximately a factor of 2 less than LC and shows that CE makes a good low-dilution separation system. Detection may still be a problem because very low sample volumes and masses are typically used in CE and the path length for on-capillary detectors is small (which affects the Φ term in eqs 2 and 3). As will be shown below, a large splitting of the sample volume is necessary to accommodate the volumes necessary for proper operation of the CE experiment. This may lead to more serious detection problems for CE in the multicolumn format. Combinations of Techniques. The five types of separation systems discussed previously and summarized in Tables 2 through 4 can be mixed in a number of ways to form interesting multidimensional systems. We can use a matrix and examine the various components of the net dilution factor formed by the product of individual column dilution factors and split ratios as embodied in eq 23. One should note that the second and higher dimension separation systems should operate at a faster rate than the first dimension so that adequate peak sampling can take place14 in the second dimension. In that regard, typically shorter and higher performance columns need to be used in the second and higher dimensions. Toward that end, a second column in Tables 2 through 4 indicates the parameters and resulting dilution factors that may be useful in the second dimension for this type of
Table 5. Dilution Factors and Split Ratios (Row Entries Are the First Dimension; Column Entries Are the Second Dimension)
separator. The values for these second dimension systems are taken from the literature for LC,14,15 GC,11 SEC,14 CE,10 and the parameters for the second dimension FFF system discussed above. There are four entries in each matrix location in Table 5. The first entry is the product of individual dilution factors for technique 1 and technique 2 and the numbers used are taken from the first columns of Tables 2-4. This number is indicative of the overall intracolumn dilution that is present through combining two techniques. The second number in each matrix location is obtained by multiplying the dilution factors in the first dimension from column 1 of Tables 2-4 by the dilution factors in the second dimension given by the second column of Tables 2-4. The second column of Tables 2-4 is more typical of the combinations which allow faster throughput and better match the first column speed. By showing both numbers together per each technique combination, one sees that in most cases but not all that a significant reduction in the column dilution takes place. This will be discussed further below. The third number is the split ratio needed to couple the first dimension technique given in column 1 in Tables 2-4 with the second dimension technique given in column 2 of Tables 2-4. This is calculated by eq 14 noting that N′ ) 3 for the numbers given in this paper. The fourth number in Table 5 in boldface is the net dilution factor that is obtained by multiplying the second number, the dilution due to column processes, by the third number, the split ratio, which gives the net dilution due to both intracolumn and intercolumn dilution. It is this number that is
most important in the assessment of the coupling of multidimensional techniques. If the split ratio is less than one, which means that the injection volume is greater than the volume of effluent, the fourth number is given as just the third number because all of the sample can be accepted without splitting. While GC can be an interesting first dimension, the condensation of gas or vapor back to the liquid state is impractical. Hence, GC is only recognized here as a second dimension separation system except where GC is also specified as the first dimension separation system. A number of different aspects of interfacing these techniques are shown by the numbers in Table 5. Perhaps the most dominant aspect of this table is that only certain specific combinations of techniques appear to offer net dilution factors below 2500. These include LC/LC, LC/SEC, SEC/LC, SEC/SEC, and the combinations where CE is a first dimension separator. In addition, FFF does not appear to be a desirable component in forming practical multidimensional separation systems. Since FFF with the current channel widths in routine use is not a fast separator, it is more suited as a first dimension system. However, the net dilution factors appear huge in this configuration. Both the column dilution factors and the split ratios appear very large. Even with a dual outlet, which would bring the first dimension dilution factor from 1340 to approximately 220, a huge net dilution still takes place. From Table 5 both SEC and CE appear to be low column dilution systems which are compatible under certain restrictions with the other systems except FFF. However, the split ratios given in Table 5 show that when CE is in the second dimension extremely large split ratios must be used to accommodate the very small injection volumes. This places limits on the utilization of CE in the second dimension and this will be discussed further below where we examine CE in the second dimension when microcolumns are used in the first dimension. SEC appears to be one of the best techniques for low dilution and for use primarily as a second dimension separator. It is a particularly good second dimension separator when used with LC in the first dimension because these techniques tend to be complimentary; i.e., the retention mechanisms of size (SEC) and hydrophobicity (RPLC) tend to be somewhat orthogonal allowing for greater separation selectivity in multiple dimensions. SEC is also better in the second dimension because it is far more convenient to run one long gradient elution experiment with LC in the first dimension as opposed to many short gradients when LC is in the second dimension. This is due to the excess time needed for equilibration which is impractical for LC in the second dimension when gradient elution is to be performed. For isocratic operation this is not important, but SEC followed by LC has higher net dilution due to the increased split ratio. It appears from Table 5 that GC as the second dimension is most difficult because the split ratios are generally quite large for GC. This is due to the low injection volumes required for GC operation. If splitless operation can be conducted where the head of the GC column in the second dimension is used for sample preconcentration as discussed previously for GC/GC, the column dilution factors in Table 5 indicate that GC makes a good second dimension separation system. This may be very difficult for liquid systems like LC/GC where large volumes come from the first Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
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dimension system. However, we will show shortly that systems like LC/GC can be improved by using a microcolumn in the first dimension so that the eluent volume is more compatible with the capillary format of GC. Other methods of coupling LC to GC, for example, using retention gaps, have recently been reviewed.37 The techniques in Table 5 which are on the matrix diagonal entries are interfaced to the same technique and do not usually offer much enhancement in selectivity. For example, SEC/SEC and CE/CE do not offer much advantage for more detailed separations, but these systems appear to offer relatively small net dilution factors. The exceptions to this are LC/LC and GC/GC. In LC/LC NPLC, RPLC, ion exchange, and other techniques which differ in retention mechanisms can be readily mixed with good success.15,38 GC/GC11 appears to be a viable technique when performed at different column temperatures so that some form of entropy difference can be exploited. In many cases it is noted from Table 5 that much smaller column dilution is exhibited when the shorter columns given in the second columns of Tables 2 through 4 are utilized. For example, in LC/LC, the use of two conventional high-performance columns of normal length gives a column dilution factor of 630. However, when a shorter and higher performance column is used in the second dimension, a factor of approximately 4.5 less zone dilution can be effected. This dilution reduction varies throughout Table 5 but in many cases is greater than 4. For cases where trace analysis is performed and detector sensitivity is an issue, these savings are quite important. In addition, the smaller length columns, as discussed in ref 14, allow faster second dimension column operation, which promotes faster first dimension column operation. It must be noted that in Table 2, where the conditions for the second dimension LC column are given, the fraction of plates lost due to finite injection volume is quite high, φ ) 0.150 in this case. This is due to the small column length used here and reflects a tradeoff with second dimension techniques between column length, efficiency, and speed. The faster desired speed suggests a smaller column length which loses efficiency due to both finite volume broadening and the smaller number of plates from a shorter column. It appears that the best way to interface any technique with CE is to use other microcolumn techniques in conjunction with CE, for example, microcolumn LC. A comparison matrix similar to Table 5 but now with microcolumn LC and SEC will be presented shortly. When GC is used as a second dimension separation system, the injection volume requirements also appear to be more compatible with capillary techniques such as microcolumn LC.39 This is because the microcolumn techniques require relatively low injection volumes and the volume scale utilized in these microcolumn techniques is somewhat incompatible with the larger column volumes exhibited in “standard bore” LC and SEC columns. To see what types of dilutions and split ratios would be present with use of microcolumns for LC and SEC, Table 6 contains typical values for microcolumn LC and microcolumn SEC. We utilize these values for the types of calculations given in Table 5 and (37) Hyo¨tyla¨inen, T; Riekkola, M.-L. J. Chromatogr., A 1998, 819, 13-24. (38) Holland, L. A.; Jorgenson, J. W. Anal. Chem. 1995, 67, 3275-3283. (39) Cortes, H. J.; Pfeiffer, C. D.; Richter, B. E. J. High Resolut. Chromatogr. 1985, 8, 469-474.
1654 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
Table 6. Summary of Conditions for the Calculation of Dilution Factors for µ-LC and µ-SEC technique µ-LC
µ-SEC
first dimension
second dimension
dc ) 250 µm dp ) 5 µm k′ ) 2 L ) 25 cm Nsep ) 10000 Vinj ) 60 nL ) 0.8
dc ) 250 µm dp ) 3 µm k′ ) 2 L ) 3 cm Nsep ) 1500 Vinj ) 25 nL ) 0.8
f ) 12.5 Vdil ) 0.750 µL VR ) 29.5 µL φ ) 0.0302
f ) 9.40 Vdil ) 0.235 µL VR ) 3.53 µL φ ) 0.0533
dc ) 250 µm dp ) 5 µm Ksec ) 0.5 L ) 100 cm Nsep ) 4000 V0 ) 17.0 µL Vp ) 20.0 µL Vinj ) 0.1 µL
dc ) 250 µm dp ) 3 µm Ksec ) 0.5 L ) 20 cm Nsep ) 1000 V0 ) 3.40 µL Vp ) 4.00 µL Vinj ) 0.1 µL
f ) 10.7 Vdil ) 1.07 µL VR ) 27.0 µL φ ) 0.00455
f ) 4.34 Vdil ) 0.434 µL VR ) 5.40 µL φ ) 0.0278
Table 7. Dilution Factors and Split Ratios When Microcolumns Are Utilized (Row Entries Are the First Dimension; Column Entries Are the Second Dimension)
present the results in Table 7. Note that FFF is not included in Table 7 because even the small w channels given in the second column of Table 3 are not really microchannels in terms of the solvent volume. Even for this case the retention volume is 6.71 mL, which is quite large.
The split ratios used for GC in the second dimension, as given in Table 7 where microcolumns are used for LC and SEC, show more compatibility as they are much smaller than those given in Table 5. For most of the entries in Table 7 but not all, the net dilution factors are smaller than those shown in Table 5. Again, smaller volumes are used in these LC and SEC columns, and these are more compatible with the smaller injection volumes needed for GC and CE, as compared with standard bore columns. In some cases, for example, LC/LC, the split ratio is slightly larger with microcolumns. This is partially due to the limited injection volume that is present with the microcolumn, although this may be just dependent on the conditions that have been chosen here. In regards to using CE as the second dimension with microcolumns, it is shown in Table 7 that the split ratios are still very large; CE is difficult to interface because most of the sample should be discarded when high resolution separation is sought. When CE is used as the first dimension, this is not a problem as the whole sample can be effectively injected into the next dimension separation system. By comparing LC/CE with CE/ LC in Table 7 it is seen that the column dilution factors are of the same order; however, the split ratios are drastically different. Electrophoresis followed by chromatography has been shown in the comprehensive mode40 to be very useful. In this case CE was not used but rather a higher throughput gel system capable of larger solute loadings was used. With CE as the first dimension separator, small sample volumes must be used to maintain high resolution. Further dilution on an already small sample mass occurs in the second and higher dimensions, placing a great burden on detection systems. Planar Geometry. Planar multidimensional separation systems such as polyacrylamide gel/isoelectric focusing slabs and two-dimensional TLC systems are common. The extension of these geometric systems to three dimensions has been discussed in the literature18 but has yet to be shown to be viable. This may change in the future due to the advent of various technologies which can image spatial slices such as confocal microscopy.41 However, for simplicity we will limit the discussion of planar separation systems to two dimensions. The theory is easily extended to the three-dimensional case. Detection for these planar systems is considered to be “on slab” with sophistication ranging from densitometry to matrix-assisted laser desorption-time-of-flight mass spectrometry (MALDI-TOF/MS), which has been shown to be useful as a detector for TLC.42,43 The theory for zone dilution in a two-dimensional geometry is easily produced under the constraint that scattering processes44 seen in slabs are not considered here. Rather, a simple linear relationship between the signal and concentration is assumed. The detection cross-section in two dimensions is the x-y plane. The multidimensional Gaussian concentration density, C(x, y), in (40) Rose, D. J.; Opiteck, G. J. Anal. Chem. 1994, 66, 2529-2536. (41) Confocal Microscopy; Wilson, T., Ed.; Academic Press: San Diego, CA, 1990. (42) Gusev, A. I.; Vasseur, O. J.; Proctor, A.; Sharkey, A. G.; Hercules, D. M. Anal. Chem. 1995, 67, 4565-4570. (43) Mehl, J. T.; Nicola, A. J.; Isbell, D. T.; Gusev, A. I.; Hercules, D. M. Am. Lab. 1998, June 30-38. (44) Huf, F. A. In Quantitative Thin-Layer Chromatography and Its Industrial Applications; Treibner, L. R., Ed.; Marcel Dekker Inc.: New York, 1987; Chapter 2, pp 17-66.
the spatial variables x and y is the product45 of the single dimension concentration density functions C(x), C(y), and C(z) such that
C(x,y) ) minj‚C(x)‚C(y)‚C(z)
(36)
Note that C(z) is the uniform density function 1/hz, where hz is the planar bed zone thickness and is assumed to be uniform throughout x and y. Using Gaussian densities for C(x) and C(y) yields
C(x,y) )
[
]
[
minj 1 -(x - jx)2 -(y - jy)2 1 ‚ exp ‚ exp hz x2πσ 2σx2 2σy2 x2πσy x
]
(37) where σx and σy are the spatial standard deviations of the zone, and jx and jy are the positions of the zone maximum. At the zone maximum
Cmax ) C(xj,yj) )
minj 1 ‚ 2πhz σx‚σy
(38)
This can be further expanded using the previously mentioned relationship σL ) L/xN to give
Cmax )
minj xNxNy ‚ 2πhz rxry
(39)
where rx and ry are the vector components of the migration distances of the zone which are measured from the zone center to the point of zone introduction (injection). In addition, Nx and Ny are the number of theoretical plates in the x and y coordinates. Equations 38 and 39 can be cast in the form of a dilution volume by analogy with eq 1:
Vdil ) 2πhzσxσy )
2πhzrxry
xNxNy
(40)
One result immediately apparent between the dilution volume in a two-dimensional planar geometry case given in eq 40 and the multidimensional column case given by eq 28 is that the planar geometry case does not require the specification of the injection volumes. This is because in the column case the injection volume into the succeeding column cannot be accommodated and a splitter is necessary. This is not present in the planar case where all of the zone is accommodated in both dimensions. In some respects this makes the two-dimensional (and three-dimensional) planar system more like a regular single column system. The zone dilution in two-dimensional TLC is evaluated by using some of the numbers which have been reported42,43 in the coupling of TLC and MALDI-TOF/MS. These are listed in Table 8, using eq 40 to obtain f and Vdil. As can be seen from Table 8, the dilution factor and the dilution volume are very low as compared to both standard bore LC given in Table 5 and the microcolumn LC results (45) Crame´r, H. Mathematical Methods of Statistics; Princeton University Press: Princeton, NJ, 1974.
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Table 8. Summary of Conditions for the Calculation of Dilution Factors for TLC hz ) 50 µm σx ) 1 mm σy ) 1 mm
Vinj ) 0.157 µL f)2 Vdil ) 0.314 µL
given in Table 7. This suggests that separations performed in the planar mode are ideal from the viewpoint of low dilution and offer a very good coupling between the first and second separation dimensions. The mechanical coupling between the separation and detection phase42,43 is not automated in this case and fraction collection is mechanical, i.e., zones need to be scraped. However, the dilution factor for a two-dimensional TLC experiment will be approximately 4 as presented in Table 8, which suggests that with sophisticated detectors such as MALDI-TOF/MS, planar separation techniques may be an important option for future multidimensional separation systems. SUMMARY OF FINDINGS A theoretical framework has been established for multidimensional separations whereby column dilution factors for each technique and split ratios can be multiplied to yield the net dilution and the multidimensional mass limit of detection. Using this simple theoretical development a number of highlights are shown. These include the calculation that shows that FFF techniques, in their present state of development, dilute sample zones far in excess of other column methods, making their use as multidimensional separation system components most difficult for easy zone detection. We have also seen that techniques on the matrix diagonal from Tables 5 and 7 show good compatibility when the retention mechanisms are sufficiently different to warrant their use. Both SEC/SEC and CE/CE appear unwarranted as does FFF/FFF because of the large dilution; however, LC/LC and GC/ GC are well suited toward compatible systems. This study has emphasized that CE is most difficult to effectively utilize as a second dimension because the injected sample zone must be very thin to maintain high plate number and high resolution. However, CE and other electrophoretic systems can make usable first dimension systems that are not as tricky to engineer. Perhaps the easiest technique to use in the context of multiple dimensions is SEC because the injection volumes are large. SEC is also not a large generator of dilution volume especially in the microcolumn format. SEC makes both a good first dimension and second dimension system. For better compatibility with CE and GC, microcolumns can be used to advantage for SEC and LC so that split ratios are smaller and easier to control to provide more versatile systems interfacing. Smaller length columns in the second dimension show less dilution and permit faster operation than standard length columns. Finally, planar techniques exhibit very low dilution and are shown to be easier to incorporate into a lower dilution system than the column methods. Column methods have a much larger choice of the types of phases and methodologies, especially considering that the stationary phase typically stays the same in both dimensions of a planar separation system. However, if detection systems such as MALDI-TOF/MS can be routinely 1656 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
utilized for multidimensional separations, chromatography and electrophoresis implemented in the planar separation mode can be expected to aid in the quest for lower net dilution and lower mass limit of detection multidimensional separation systems. ACKNOWLEDGMENT I thank Nancy Schure, of Kroungold Analytical, Blue Bell, Pennsylvania, for editorial assistance, and Robert Murphy of Isis Pharmaceuticals, Carlsbad, California, and Dan Dohmeier of Rohm and Haas for many helpful suggestions. SYMBOLS A
cross-sectional area of a capillary
An
noise amplitude
b
channel breadth in FFF channels
C(t)
concentration as a function of time
C(x), C(y), C(z) concentration as a function of the spatial coordinate x, y, or z Cmax, Ci,max
concentration at the peak maximum
Cinj, Ci,inj
injection concentration
q q Cmax , CM,max
minimum detectable concentration at the peak maximum
q C0,inj
q initial injection concentration resulting in CM,max at the detector
C h
average concentration in the sample loop
D
diffusion coefficient
d
particle diameter in FFF
dc
column diameter
F
flow rate
f, fi
dilution factor equal to Vdil/Vinj
hf
average value of f(t)
f(t)
general time-varying function
H
plate height
hz
planar bed zone height or thickness in the z coordinate
KSEC
equilibrium constant used in SEC (0 e KSEC e 1)
k′
capacity factor
kB
Boltzmann’s constant
L
column or channel length
Linj
spatial length of injected zone
M
number of separation dimensions
Mw
molecular weight mass of injected solute
minj mqinj,
q m0,inj
minimum detectable mass of injected solute (limit of detection)
N, Nx, Ny
number of theoretical plates
Ninj
number of plates due to finite volume injection
Nsep
number of plates from separation processes
NT
total number of plates including separation and injection
N′
number of samples taken across a peak
R
retention ratio equal to t0/t′
rx, ry
zone maxima distance from injection point in the x and y coordinates
z
the axial separation coordinate of a column or channel
Sd
detector sensitivity (amplitude per unit concentration)
total fraction of column volume external to particle surface
Si
split ratio in the ith dimension
b
T
absolute temperature
fraction of the column volume contained in the interstitial region
t
time
p
t0
void time
fraction of the particle volume contained in the pore
ts
sampling time
η
viscosity
t′
elution time
λ
tq
time between peak maximum and commencement of sampling
nondimensional mean layer thickness used in FFF theory
σ
time-based Gaussian standard deviation
σinj
standard deviation in length units due to injection
σL, σi,L
length-based Gaussian standard deviation
σT
total length-based Gaussian standard deviation
σv
volume-based Gaussian standard deviation
σx, σy
length-based standard deviation in the x or y coordinate
φ
fraction of separation plates lost due to injection process
χ
nondimensional nonequilibrium coefficient used in FFF theory
Γ
nondimensional ratio of tq/σ
Θ, Θi
C h /Cmax
Φ
the ratio An/Sd
u
general variable
V
volume
Vdil, Vi,dil
dilution volume
VTdil
total dilution volume for an M column system
Ve
effective volume of the column under limiting conditions
Vi,s
volume of eluent from column i per sampling period
Vinj, Vi,inj
injection volume
Vp
internal pore volume in SEC
VR
retention volume
V0
void volume or SEC interstitial pore volume
〈v〉
average fluid velocity
w
channel width in FFF channels
x
separation coordinate
jx
position of zone maxima in the x coordinate
y
separation coordinate
Received for review October 15, 1998. Accepted February 4, 1999.
jy
position of zone maxima in the y coordinate
AC981128Q
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