Geometry-Independent Plate Height Representation Methods for the

May 3, 2005 - These KP numbers are much more informative than the Hmin, u0,opt, and Kv data traditionally employed to quantify the performance of LC s...
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Anal. Chem. 2005, 77, 4058-4070

Geometry-Independent Plate Height Representation Methods for the Direct Comparison of the Kinetic Performance of LC Supports with a Different Size or Morphology Gert Desmet,* David Clicq, and Piotr Gzil

Department of Chemical Engineering (CHIS-TW), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium

The advantages of representing experimental plate height data as a plot of Kv/u02 or H2/Kv versus Kv/(Hu0) instead of as H versus u0 are discussed (Kv)column permeability). Multiplying the values on both axes by the ratio of a reference pressure drop and mobile-phase viscosity, the obtained plots directly yield the kinetic performance limits of the tested support structure, without any need for further numerical optimization. Directly showing the range of plate numbers or analysis times wherein the tested support geometry can yield faster separations or produce more plates than another support type, such kinetic plots are ideally suited to compare the performance of differently shaped or sized LC supports. The approach hence obviates the need for a common reference length, which is a clear problem if it is attempted to compare differently shaped supports on the basis of their flow resistance O and reduced plate height h. It is also shown how an MS Excel template file, only requiring the user to paste the column permeability Kv and a series of experimental (u0, H) data, can be used to automatically establish a series of so-called kinetic performance (KP) numbers, which can be used to completely describe the performance characteristics of the considered support. The advantages of the proposed data representation methods are demonstrated by applying them to several recent literature plate height data sets, showing that the obtained kinetic plots directly visualize the range of plate numbers where new approaches such as ultra-highpressure HPLC or the use of open-porous silica monoliths can be expected to provide a substantial gain and where not. The data analysis also showed that the most generally relevant KP numbers are Nopt (the plate number for which the support achieves its best analysis time/pressure cost ratio), topt (the time needed to obtain Nopt plates), and t1K (the time needed to generate 1000 or 1 kilo of theoretical plates). These KP numbers are much more informative than the Hmin, u0,opt, and Kv data traditionally employed to quantify the performance of LC supports. The recent years have witnessed the development of new LC support formats such as polymer and silica monoliths or COMOSS columns,1-5 introducing solid-phase and through-pore morphol* To whom correspondence should be addressed. Tel.: +32.(0)2.629.32.51. Fax: +32.(0)2.629.32.48. E-mail: [email protected].

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ogies that are totally different from those encountered in the packed bed of spheres. One of the problems emanating from the research on these new support types is that it has turned out to be quite difficult to find a fair and simple method to compare their separation performances.6-8 As long as only packed bed columns were used, columns could be compared on the basis of their plate heights near the minimum and in the C-term-dominated range of the van Deemter curve. However, with the introduction of polymer and silica monolith columns.9-11 typically producing a significantly smaller flow resistance than the packed bed of spheres, it no longer suffices to compare only plate heights, because then the information on the flow resistance and the concomitant possibility to apply larger flow velocities or use larger column lengths is not accounted for. Switching to reduced plate heights, another very popular approach for the comparison of different support formats,1,6,9-10 this problem remains. In this case, a second problem arises, because it seems to be difficult (if not impossible) to exactly identify a general reference length with which the plate height curves of, for example, a packed bed and a silica monolith column can be made to coincide on the same reduced plate height curve. The domain size, defined as the sum of the pore and skeleton (or particle) size,9 currently seems the best possible guess for this reference length, but obviously still is a too simple measure to account for all the shape differences.12 It is furthermore not easy to define and measure in real columns and packings. Chromatographers desiring to speed up their analysis and wondering whether they should switch to a new support format hence have to evaluate at least two different performance criteria: the minimal plate height Hmin and the column perme(1) Tanaka, N.; Kobayashi, H.; Nakanishi, K.; Minakuchi, H.; Ishizuka, N. Anal. Chem. 2001, 420-429. (2) Majors, R. E. LC-GC Eur. 2003, 16 (6a), 8-13. (3) Rozing, G. LC-GC Eur. 2003, 16 (6a), 14-19. (4) Svec, F. LC-GC Eur. 2003, 16 (6a), 24-28. (5) He, B.; Tait, N.; Regnier, F. E. Anal. Chem. 1998, 70, 3790. (6) Knox, J. H. J. Chromatogr., A 2002, 960, 7-18. (7) Leinweber, F. C.; Lubda, D.; Cabrera, K.; Tallarek, U. Anal. Chem. 2002, 74, 2470-2477. (8) Leinweber, F. C.; Tallarek, U. J. Chromatogr., A 2003, 1006, 207-228. (9) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. J. Chromatogr., A 1997, 762, 135-146. (10) Minakuchi, H.; Nakanishi, K.; Soga, N.; Ishizuka, N.; Tanaka, N. J. Chromatogr., A 1998, 797, 121-131. (11) Svec, F.; Frechet, J. M. Anal. Chem. 1992, 64, 820-822. (12) Gzil, P.; Vervoort, N.; Baron, G. V.; Desmet, G. J. Sep. Sci. 2004, 27, 887896. 10.1021/ac050160z CCC: $30.25

© 2005 American Chemical Society Published on Web 05/03/2005

ability Kv (or hmin and φ in reduced measures). The relative importance of both measures is, however, difficult to assess. Questions such as “is a two-fold reduction of φ as useful as a twofold reduction in h?” have no straightforward quantitative answer. One of the most frequently used approaches to resolve this problem is to combine both measures into the separation impedance number E0, originally defined by Bristow and Knox13 as

E0H2/Kv ) h2φ

(1)

and generally considered as the ultimate figure of merit of LC supports.14,15 If columns with a different morphology are to be compared, the E0 number also elegantly circumvents the need for a common characteristic reference length. It has therefore found widespread use as the main performance characteristic for the comparison of monolithic with packed bed columns.16-18 Being a dimensionless quantity, the E0 number however does not give any real information on the speed of analysis. The easiest way to understand this is by noting that packed beds with, for example, dp ) 2-, 5-, and 10-µm particles all have the same E0 number (if all equally well packed), but the time they need to produce, for example, 30 000 theoretical plates is totally different. In the present paper, we want to make a plea for an alternative plate height data representation method, the so-called kinetic plot method, which automatically yields the relevant performance characteristics for two of the most important chromatographic optimization problems, i.e., achieving a maximal number of plates in a given analysis time and minimizing the analysis time needed to achieve a given number of plates. Although not needed to apply and understand the proposed data representation method, the remainder of this introduction is devoted to the difference between self-similar and non-self-similar structures. This part is needed to understand some of the properties of the obtained plots (e.g., to discuss the position of the Knox and Saleem limit in Figure 3). Self-similar structures are structures having a different size, but which can be made to perfectly overlap by doing a thought experiment wherein the characteristic dimensions describing their morphology are proportionally varied. Put otherwise, the ratios of the different characteristic measures (determined from SEMs or laser scanning confocal microscopy pictures) needed to describe their morphology should be independent of their absolute size, as was approximately the case in the study in ref 10. Silica monoliths with the same porosity or packed beds of spherical particles can, for example, be expected to be self-similar. Polymer or silica monoliths with a different external porosity on the other hand are not self-similar, because their skeleton cannot be made to overlap by simply rescaling its size. Since Giddings’ analysis of self-similar systems and his introduction of the reduced plate height concept,19 (13) Bristow, P. A.; Knox, J. H. Chromatographia 1977, 10, 279-289. (14) Poppe, H. J. Chromatogr., A 1997, 778, 3-21. (15) Knox, J. H. J. Chromatogr. Sci. 1980, 18, 454-461. (16) Tanaka, N.; Kobayashi, H.; Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Hosoya, K.; Ikegami, T. J. Chromatogr., A 2002, 965, 35-49. (17) Motokawa, M.; Kobayashi, H.; Ishizuka, N.; Minakuchi, H.; Nakanishi, K.; Jinnai, H.; Hosoya, K.; Ikegami, T.; Tanaka, N J. Chromatogr. A 2002, 961, 53-63. (18) Ishizuka, N.; Kobayashi, H.; Minakuchi, H.; Nakanishi, K.; Hirao, K.; Hosoya, K.; Ikegami, T.; Tanaka, N. J. Chromatogr., A 2002, 960, 85-96.

it is well-established that all members of the same self-similar structures group (SSG) can be expected to have identical a, b, and c constants in the reduced Knox equation, provided this reduction is based on any of the characteristic dimensions (dref) of their geometry:

h ) aν0n + b/ν0 + cν0

(2)

It can hence also be expected that all members of the same SSG have the same hmin, νopt, and φ, and as a consequence, also the same E0,min number. If members of the same SSG have a different E0,min number, this difference can in fact be used to identify differences in support homogeneity (i.e., packing quality in the case of packed bed columns). For supports belonging to different SSGs, the E0,min number can be expected to be intrinsically different because of the different shape factors in the equations describing the flow and mass transfer. Below, the reader is introduced into all the mathematical foundations of the presently proposed kinetic plot method. More practically oriented readers who are not interested in these subtleties can suffice by only considering eqs 6-8 in combination with Figure 1 and can then immediately proceed to the Application to Some Typical Literature Data Set section. KINETIC PLOT METHOD Method Description and Validation. Based on the wellestablished expression for the analysis time in a chromatographic system

tR ) t0(1 + k′)

with

t0 ) NH/u0

(3)

the general pressure drop-limited kinetic optimization problem15,20-23 accounts for the fact that the unretained peak velocity u0 cannot be freely selected but has to respect the pressure drop equation, which can conveniently be written as15

u0H )

∆P Kv η N

(4)

Equations 3 and 4 are coupled via the van Deemter equation, linking the plate height to the velocity of the mobile phase u0, and reformulated by Knox6 as

H ) Au0n + B/u0 + Cu0

(5)

In the conventional kinetic optimization approach, established by Poppe,14,24 a series of different N values is selected, and the maximal allowable u0 velocity is subsequently calculated for each of these values by simultaneously solving eqs 4 and 5 using a numerical procedure wherein u0 is advanced in small steps until (19) Giddings, J. C. Dynamics of Chromatography; Marcel Dekker: New York, 1965; Part I. (20) Giddings, J. C. Anal. Chem. 1965, 37, 60-63. (21) Knox, J. H.; Saleem, M. J. Chromatogr. Sci. 1969, 7, 614-622. (22) Guiochon, G. Anal. Chem. 1981, 53, 1318-1325. (23) Katz, E.; Ogan, K. L.; Scott, R. P. W. J. Chromatogr. 1984, 289, 65-83. (24) Popovici, S. T.; Schoenmakers, P., Poppe-plots for size Exclusion Chromatography, J. Chromatogr. A. Web-published.

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eq 4 is satisfied. The thus obtained values for u0 and H are then inserted into eq 3 to yield the t0 or tR value corresponding to the selected N. To speed up the calculations, a bisection method can be used.14,24 The fact that the experimental plate height data first need to be fitted to obtain the A, B, and C parameters in eq 5, added to the need for a numerical optimization algorithm to simultaneously solve eqs 4 and 5, makes this method rather cumbersome and certainly not straightforward to apply for the nonmathematical expert. The only exception for which no numerical solution procedure is needed is the special case of eq 5, of n ) 0. Considering that a description of the kinetic performance of an LC support does not necessarily require that the minimal analysis time is determined for a series of exactly specified N values, but that it suffices to do this for a number of well-spread N values, the number of plates N can however also be simply removed from the problem by rewriting eq 4 as

N)

( )[ ] ∆P Kv η u0H

(6)

exp

and inserting this into eq 3, yielding

t0 )

( )[ ] ∆P Kv η u2 0

(7)

exp

Equations 6 and 7 show that recombining the experimental u0, Kv, and H values into the two groups between the brackets and multiplying them with the ratio of two scaling factors (∆P and η) directly yields a plot of the analysis time t0 versus N. This data transformation method is illustrated in Figure 1a and b, where the method has been applied to two series of packed bed data obtained by using eq 5 with n ) 1/3 and by filling in some typical, self-chosen A, B, and C values. The latter was needed to obtain a set of plate height data with exactly known A, B, and C values to compare the current method with the numerical Poppe method described in refs 14 and 16 without any fitting error. Feeding the thus-obtained plate height data to an self-written numerical optimization algorithm (written in Fortran 90 programming code) performing the kinetic optimization procedure described in refs 14 and 24 yielded the two red curves shown in Figure 1b. The agreement of these numerically calculated curves with the (N,t0) data obtained by simply carrying out the multiplications and divisions in eqs 6 and 7 clearly demonstrates that the latter also yield the fully optimized plate number/analysis time relationship, but without the need for any numerical calculation or optimization routine. This is in fact not surprising, since the physical meaning of eqs 6 and 7 is that they transform the (u0,) data, which are obtained in a column with a given length and a given pressure drop into a projection of the plate number and the corresponding t0 time, which would be obtained if the same support would be used in a column taken exactly long enough to yield the maximal allowable inlet pressure if the given u0 velocity would be applied. The convention adopted in eqs 6 and 7 is that all experimental values are grouped between the brackets, whereas the parentheses combine all the normalization factors. The most important of the latter is the pressure drop ∆P. To compare the intrinsic kinetic 4060

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performance, all systems should be evaluated for the same ∆P, independently of the actually applied experimental pressure drop. An obvious choice for ∆P would be the normal operating limit of commercial HPLC equipment, which is typically 200 or 400 bar. The viscosity η can also be treated as a normalization variable. Using a single reference viscosity for all the data series represented in the same plot allows us to compensate for differences in bed permeability originating from the use of different mobile phases. This is, however, not always a fair approach. For example, performing reversed-phase separations in highly open-porous systems, the creation of a sufficiently large retention factor will anyhow require the application of mobile phases with a larger water percentage and hence a larger viscosity. To account for this fact, it can also be preferred to use the viscosity of the actually used mobile phase for each individual data series. In this case, the η value should also be included between the brackets encompassing the experimental parameters. Convenient y- and x-Axis Modifications. A drawback of the (N,t0) plots is that their y scale usually runs over several orders of magnitude (Figure 1b), especially if supports with strongly different permeabilities are to be compared. This hinders an accurate readout of the graphs. To enable a more accurate graphical comparison, it is convenient to divide the t0 expression in eq 7 by the square of the N expression in eq 6, yielding

N)

( )[ ] ∆P Kv η u0H

(8)

exp

Equation 8 shows that the t0/N2 ratio is, except for the proportionality constant (η/∆P), equal to the E0 number originally defined by Bristow and Knox.13 The E0 number can hence be considered as a dimensionless analysis time. A plot of E0 ) H2/ Kv versus N (Figure 1d) therefore displays exactly the same tye of information as the (N,t0) plot in Figure 1b: both plots directly show the crossover point between the range of N values wherein the one packing can be expected to yield faster separations than the other, and the y scale difference is in both cases a direct measure for the difference in t0 time between both packing types. The enlarged y scale in the (N,E0) plot, however, allows for a more accurate assessment of this difference. Comparing Figure 1c and d, it can be concluded that an (N,E0) plot is much more informative than the traditionally used plots of E0 versus u0 or ν0, which do not yield any directly assessable kinetic information. Because the influence of the pressure is canceled out in its definition, the E0 number does not allow comparing systems used at a different pressure, as is, for example, needed to investigate the potential advantage the ultra-high-pressure HPLC (UHPLC) work of Jorgenson and co-workers. In this case, the dimensional t0/N2 ratio should be put on the y axis instead of the E0 number. Since this modification only involves a multiplication of the ordinate with a known scaling factor (cf. the η/∆P ratio in eq 8), it should, however, be clear that the thus-obtained (N,t0/N2) curves run perfectly parallel with the (N,E0) curves. The latter can also be witnessed from the use of the double y axis in Figure 1d. One of the major differences between the traditional (u0,H) plot in Figure 1a and the kinetic plots in Figure 1b and d is that the B-term- and the C-term-dominated range have switched places. This is, however, just a matter of habit, and if desired, the x axis

Figure 1. Transition from the conventional H versus u0 plate height representation method (a) to the t0 versus N plot (b) obtained by representing the same data sets as a plot of t0 ) (∆P/η)[Kv/u02] versus N ) (∆P/η)[Kv/(Hu0)]. The two data sets were obtained by taking two typical15 packed bed literature values: packed bed 1, a ) 0.5 and φ ) 833 (b) and packed bed 2, a ) 0.8 and φ ) 500 (O). Other employed parameters: n ) 1/3, b ) 2, c ) 0.1, dp ) 3 µm, Dm ) 10-9 m2/s, η ) 10-3Pa‚s, and ∆P ) 400 bar. Also shown is the transition of the conventionally used E0 versus ν0 plot (c) to the E0 versus N plots which can be obtained by putting N ) (∆P/η)[Kv/(Hu0)] on the x axis: normal order (d) and inverted order (e). The red lines in (b) and (d) were obtained by using the traditional optimization method,14 i.e., fitting the a, b, and c parameters of eq 2 to the experimental data points and then using eqs 1-4 in a numerical optimization routine to determine the minimal t0 time for each considered N. The dashed lines added to (d) represent the cases of constant t0 times.

can simply be inverted to keep the same left-to-right order of the B- and C-term-dominated regimes as in the (u0,H) plots. This only requires one click in the scale window of the axis format function in MS Excel. As can be noted from Figure 1e, the thus-obtained (N,E0) and (N,t0/N2) plots display a good visual resemblance to the traditional van Deemter plot. Given that this resemblance can contribute to the general acceptance of the kinetic plot approach, the inverse N axis plot format has been maintained in the remainder of the text.

Nopt Point. Apart from their enlarged y scale, the (N,E0) or (N,t0/N2) plots have the additional advantage that the minimum of the van Deemter curve is directly visualized. This follows directly from eq 1, showing that E0 ∼ H2, hence implying that the E0 curve reaches its minimum at the same mobile-phase velocity as the H curve. As demonstrated already many years ago by Giddings, Knox, and many others,14,20-23 this minimum takes a special position in the kinetic performance of a given support. It determines, for example, the optimal particle or support feature Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

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size to obtain a given number of plates in the shortest time. In the present approach, the Hmin point translates into an optimal plate number Nopt, which, according to Figure 1d, can be considered as the plate number for which the support reaches its best possible kinetic performance/pressure cost ratio. This Nopt value can not only be determined from an (N,E0) plot but can also be directly calculated by filling in the experimentally obtained u0,opt and Hmin values into eq 6, yielding

Nopt )

[

]

Kv ∆P η u0,optHmin

(9)

Other Kinetic Plots. Without any significant additional effort, the (N,t0) data series yielded by eqs 6 and 7 can be further transformed to yield all other relevant kinetic performance data. Simply reversing the t0 and N axis, for example, directly yields a plot of the maximal number of plates that can be achieved in a given t0 time. Multiplying t0 with (1 + k′), using the k′ value of the actually measured component directly yields a plot of the actual analysis time tR versus N. Dividing the tR or t0 data columns by the N data and plotting this ratio versus N, the so-called Poppe plots14,25 are obtained. Replacing t0 by tR in eq 8 yields the ratio of tR/N2. Multiplying this quantity with the scaling factor (η/∆P), a series of tR-based separation impedances ER is obtained in the same way as the t0-based E0 numbers in eq 8. With another minor adjustment, it is also possible to rule out the resolution difference between experiments performed at different retention factors. For this purpose, it simply suffices to multiply the N values obtained via eq 6 with the factor [k′/(1 + k′)]2, yielding the effective number of plates Neff:26

Neff ) N

k′2 (1 + k′)2

(10)

Plotting then tR versus Neff, the kinetic data are compared on the basis of the same resolution. Respectively replacing t0 and N by tR and Neff in eq 8 yields the Neff-based separation impedance Eeff, defined as Eeff ) tR/Neff2(η/∆P), it can easily be verified that Eeff is linked to E0 via

Eeff ) E0(1 + k′)5/k′4

(11)

Plotting Eeff versus Neff then yields the k′-corrected variant of the E0 plot in Figure 1d-e. An important advantage of the Neff-based plots is that they directly reveal the value of the optimal phase retention factor k′opt. To demonstrate this, a series of synthetic plate height data has been constructed using eq 5 and the explicit c-term constant expression established by Katz et al.27 for packed beds:

c)

0.37 + 4.69k′ + 4.04k′ 2 24(1 + k′)2

(12)

Whereas the resulting (u0,H) curves shown in Figure 2a display (25) Burden, R. L.; Faires, J. D. Numerical Analysis; Brooks/Cole: Pacific Grove, CA, 2001.

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a simple monotonic increase of H with k′, the (Neff,Eeff) curves obtained after applying the transformations described in eqs 6, 7, and 10, 11 immediately show that the relation between the analysis time and the separation resolution goes through a minimum around k′ ) 3 (Figure 2b). The Neff-based kinetic plots hence directly yield a graphical solution for one of the classical problems in LC column optimization, i.e., finding the k′ value yielding the fastest critical pair analysis.23,28 The availability of such a graphical method is very useful, because there exist no analytical solutions for this problem, except for the cases of eq 5 wherein either A ) 0 (open-tubular systems) or n ) 1 (original van Deemter plate height expression). Noting the poor performance of the k′ ) 1 case in Figure 2b, it should be obvious that the Neff-based kinetic plots are also perfectly suited to put the kinetic performance of systems with a small retention or dynamic adsorption capacity (open-tubular LC, highly open-porous monoliths, packed beds with nonporous particles) into a more rightful perspective. It is also straightforward to put the column length L on the y axis. Using eq 6, the column length L can simply be obtained by calculating L ) NH ) ∆P/η(Kv/u0) for each data point. The resulting (N,L) plots are very useful because they directly allow one to assess whether the data couples in the kinetic plots do not correspond to impractically small or large column lengths. With the known (N,t0) data couples, and selecting the largest practically possible retention factor (k′), it is also very straightforward to transform the N values into a series of isocratic peak capacities (np):26

np ) 1 + N1/2/4 ln(1 + k′)

(13)

Plotting the obtained np values versus tR then directly yields the maximal isocratic peak capacity of a given support as a function of the required analysis time, another very useful plot. A freely usable MS Excel template file, automatically establishing most of the above cited kinetic performance plots, and only requiring input of a value for the column permeability Kv and a table of experimental (u0,H) data, has been posted on the Web.29 All researchers active in the field of LC support and column development are kindly invited to use the file to analyze their column performance data and to return their comments. USE OF FITTING PROCEDURES The different kinetic plots discussed above can be established and analyzed without the need for any plate height model fitting. In some instances, for example, to make accurate inter- and extrapolations, it is however advantageous to represent the experimental data by a mathematical expression. For (u0,H) data, this mathematical expression is usually obtained by a numerical fit with eq 5, or any of the other existing plate height equations, such as, for example, the generalized Giddings equation:

H)

Au0n 1 + Du0

m

+

B + Cu0 u0

(14)

(26) Cazes, J.; Scott, R. P. W. Chromatography Theory; Marcel Dekker Inc.: New York, 2002. (27) Katz, E. D.; Ogan, K.; Scott, R. P. W. J. Chromatogr. Libr. 1985, 32, 403434.

Figure 2. Transition from the conventional (u0,H) curves (a) to the (Neff,Eeff) curves (b) obtained from eqs 6, 7, and 10 for a series of synthetic plate height data obtained by putting a ) 0.5 and b ) 2 in eq 2. The c-term constant was calculated on the basis of eq 12 for six different values of k′ (k′ ) 1, 4; 2, b; 3, 0; 5, 2; 7, O; and 9, 9). Other parameter values: dp ) 3 µm, φ ) 700, Dm ) 10-9 m2/s, and η ) 10-3Pa‚s.

Filling in the appropriate values for D, n, and m, eq 14 encompasses nearly all existing plate height expressions used in the literature, including eq 5. The posted template file 29 automatically fits eqs 4 and 14 to the original experimental (u0,H) data as well as to the transformed (N,t0) data using the built-in solver procedures of MS Excel. It was our experience that the (N,t0) data fittings usually yield model parameter constants slightly different from the (u0,H) data fittings. Since the template file29 focuses on mapping kinetic performances, all parameter calculations in the file are based on the plate height model constants coming out of the (N,t0) fitting. The (u0,H) fitting is only given as supplementary information. In the template file, four different models are simultaneously fitted and compared: two Knox model variants (fixed n case and free n case) and two extended Giddings model variants (fixed n ) m case and free n ) m case). Comparing the resulting fits for a large collection of different plate height data, it was noted that the four tested models are usually all capable of producing a sufficiently accurate (N,t0) fit in the N range covered by the experimental data. Outside this range, the different models, however, tend to display a totally different extrapolation behavior, especially in the small-N range. In the large-N range, extrapolation differences only occur if the experimental plate height values are not collected sufficiently deep into the B-termdominated range of the van Deemter curve. In the small-N range, (28) Scott, R. P. W., J. Chromatogr. 1990, 517, 297-304. (29) http://wwwtw.vub.ac.be/chis/download.

the differences in extrapolation behavior are caused by the different A-term expressions in the different models. In supports with a small A-term contribution, all models predict a linear extrapolation in the large u0 range () range of small N) and hence all coincide very well. For systems with a strong A-term contribution on the other hand, the different models tend to produce a totally different extrapolation into the small-N range. This extrapolation problem is discussed further below. IMPORTANT CURVE CHARACTERISTICS Determination of the Knox and Saleem Limit, the van Deemter Curve Minimum, and the Optimal and Maximal Plate Number (Nopt and Nmax). One of the most important kinetic performance characteristics is the Knox and Saleem limit.14,21 This limit is linked to the van Deemter curve minimum and is obtained by rewriting eq 7 for the special case of H ) Hmin, yielding 2

t0 ) N 2

η Hmin η ) N2 E ∆P Kv ∆P o,min

(15)

Considering then that according to Giddings’ dimensionless number analysis19 all members of the same SSG have the same E0,min number (provided they have a similar homogeneity), eq 15 describes a straight-line relationship between t0 and N2. This is Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

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that the dimensionless parameters hmin, ν0,opt, and φ are should be identical for each member of a SSG, it follows from eq 16 that Nopt is directly proportional to dref2. Hence, if for example a monolithic silica column with a skeleton size of ds ) 2 µm yields an Nopt value of 400 000 plates, this Nopt value would reduce to Nopt ) 100 000 if a monolith with similar porosity, similar poreto-skeleton size ratio, and similar homogeneity could be produced with a skeleton size of ds ) 1 µm. Since this support then has its Nopt at N ) 100 000, it would allow us to perform the N ) 100 000 plate separation in the shortest possible separation time, shorter than with any other silica monolith of the same SSG. Although less important from a kinetic point of view (columns are seldom used in the B-term regime), the vertical asymptote of the kinetic plots obviously is one of their most eye-catching characteristics. This asymptote corresponds to the maximal number of plates (Nmax) that can ever be achieved with the given support. Nmax can easily be calculated using eq 6 and by noting that the (Hu) product appearing in the denominator of this equation is simply equal to B in the small velocity limit, such that

Nmax )

Figure 3. Position of the Knox and Saleem limit line and the Nopt points in the (N,t0) plot (a) and the (N,E0) plot (b) of three differently sized (dp ) 1 µm, b; dp ) 3 µm, and dp ) 5 µm, 9) support systems belonging to the same SSG, i.e., having the same a, b, and c constant values in eq 2. Parameter values: a ) 0.5, b ) 2, c ) 0.095, φ ) 700, η ) 10-3 Pa‚s, and Dm ) 10-9 m2/s.

the Knox and Saleem limit,14,22 known to connect the optimal working points of all the differently sized members of the same SSG. In the logarithmic version of the (N,t0) plot, eq 15 corresponds to a straight line with a slope of 2. The Knox and Saleem limit line can hence easily be found by translating a straight line with slope 2 until it touches the experimental (N,t0) curve (Figure 3a). This touching point then yields the Nopt, Hmin, and uopt values of the tested support. In the (N,E0) plots, the Knox and Saleem limit is simply found by drawing a straight horizontal line passing through the curve minimum (Figure 3b). As is illustrated in Figure 3, structures of the same SSG touch the same Knox and Saleem limit line but do this at different Nopt values, depending on their size. At its own Nopt value, each member of a given SSG performs better than any other member of the same SSG. If another number of plates is to be achieved, a differently sized member of the same SSG should be selected to yield the shortest possible analysis time. To select the optimal size, it should first be noted that eq 9 can be rewritten in terms of the reduced variables h, ν0, and φ by introducing a given reference length dref, describing one of the characteristic geometric features of the support

Nopt ) dref2

( )[ ∆P ηDm

1 hminν0,optφ

]

(16)

exp

Considering then that the theory of self-similar structures dictates 4064

Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

[]

∆P Kv η B

exp

(17)

This expression, also already well-established for a very long period,20,21 allows one to calculate the B-term constant directly from the known position of the experimental Nmax value. Inversely, eq 17 can be used to directly predict Nmax for a given support, as soon as its B and Kv values are known. Influence of Inlet Pressure. With the groundbreaking work of Jorgenson’s group on UHPLC,30-33 it has become very important to be able to rapidly assess how a given support structure would perform under different pressure gradient conditions. Figure 4 shows how a single experimental (u0,H) data series (taken from ref 32) is transformed into different (N,t0) curves when the reference pressure gradient is shifted from ∆P ) 40 bar to ∆P ) 4000 bar. This exercise clearly illustrates how the Nopt and Nmax values of a given support type increase in a linearly proportional way with the applied ∆P. In fact, this simple linear translation holds for all the experimental points, as can be understood from eqs 6 and 7 and as can clearly be noted from Figure 4, where all the individual experimental points obviously are translated along a parallel bundle of straight lines with slope of 1 (see dashed lines). The reader should, however, note that any rescaling made by changing ∆P in eqs 6 and 7 is implicitly based on the assumption that all experimental variables (plate heights, mobile-phase viscosity, retention factor, diffusivity) are independent of the applied inlet pressure. This assumption is known to be doubtful above 400 bar, cf. the reported changes in viscosity, diffusivity, adsorption equilibria, etc., occurring at higher pressures.32,33 Hence, kinetic data obtained from experimental H values obtained at or below 400 bar and then recalculated by assuming a ∆P value larger than (30) MacNair, J. E.; Lewis, K. C.; Jorgenson, J. W. Anal. Chem. 1997, 69, 983989. (31) MacNair, J. E.; Patel, K. D.; Jorgenson, J. W. Anal. Chem. 1999, 71, 700708. (32) Jerkovich, A. D.; Mellors, J. S.; Jorgenson, J. W. LC-GC Eur. 2003, 16 (6a), 20-23. (33) Patel, K. D.; Jerkovich, A. D.; Link, J. C.; Jorgenson, J. W. Anal. Chem. 2004, 76, 5777-5786.

Figure 4. Influence of ∆P on the (N,t0) plot established on the basis of dp ) 1 µm nonporous particle packed bed data obtained in a recent UHPLC study32 (sample component ) catechol with k′ ) 0.3). The dashed straight lines represent the trajectory of the individual experimental points under the influence of the varying ∆P (40 bar, b; 400 bar, 9; 4000 bar, 2; 200 bar, O; 1000 bar, 0; 2000 bar, 4).

400 bar should only be considered as a first estimate and certainly need to be experimentally verified under ultra-high-pressure conditions. Given the need for a single universal reporting standard, it is proposed to establish all kinetic plots and numbers for the standard reference case of 400 bar, even for those data obtained under ultra-high-pressure conditions. In the latter case, there is no real rescaling problem, because a downscaling of the pressure can only lead to an underestimation of the analysis speed and not to an overestimation. Kinetic Performance (KP) Numbers and Their Automated Determination. The exact position of a given curve in a kinetic plot obviously is not easy to retain or to communicate. A straightforward solution to this problem consists of reducing the total amount of information contained in the curves into a limited set of characteristic numbers. Obvious characteristic numbers are the topt and Nopt numbers (cf. Figure 3a). Considering that some of the new open-porous support types have Nopt values in the N > 1 000 000 range (cf. Figure 6 further on), these numbers are, however, not always practically relevant. It is therefore proposed to also determine the t0 or tR values needed to yield N ) 1000, N ) 10 000, N ) 50 000, etc., theoretical plates (Figure 5). For the simplicity of writing, it is proposed to use the “kilo” notation and to denote these values by t1k, t10k, t50k, etc. To denote the differences between the t0 and the tR time, it is obvious to add a “0” or “R” subscript to the employed symbols. For the case wherein the separation resolution is reported in terms of the effective plate number Neff, it is proposed to use the “eff” abbreviation in the subscript, such that the times needed to produce 1000, 10 000, 50 000, etc., effective plates would be denoted as teff,1K, teff,10K, teff,50K, etc. Since reporting a t0 time is futile in the case of effective plate numbers, it is not needed to add an “R” subscript to the teff symbol. The reciprocal kinetic plots (x axis ) time, y axis ) plate number) can equivalently be characterized by the number of plates which can be generated in 1, 10, 100, 1000, etc., seconds. These values could be represented using N1, N10, N100, N1000, etc., if referring to theoretical plates and using Neff,1, Neff,10, Neff,100, Neff,1000, etc., for the case of effective plates. Using eq 13, the Nt and tN number series can also easily be extended with a series of peak capacity

Figure 5. Illustration of the definition of some tN and tN* numbers for the packed bed 1 data of Figure 1. The boldface line represents the linear extrapolation line obtained on the basis of the most leftward situated experimental data point (Nmin point) via eq 18.

numbers, denoted as np,t, and yielding the number of peaks a given system can be expected to resolve isocratically in t seconds. If considering (N,E) curves, the tN numbers should be replaced by EN numbers. These can be defined fully analogous to the tN numbers, such that, for example, the E0,10k value gives the t0-based separation impedance for a separation requiring 10 000 theoretical plates. The ensemble of the above-defined numbers can generally be referred to as the KP numbers of a given support. These KP numbers can be determined by direct visual inspection of the different kinetic plots, but it is of course more convenient and accurate to determine them automatically. Although not exceedingly difficult, this requires some computer programming skills. To offer the automatic readout possibility to all interested researchers, the posted MS Excel template file29 contains a “macro button” which directly yields all the relevant KP numbers. The numerical routines activated by this button comprise nothing else but the standard solver function of MS Excel. They are used to seek the value of u0 for which eqs 6 and 7 are satisfied for the given value of t or N. This solver operation is repeated for a whole series of typical t and N values in one single run. The accuracy of the KP number determination obviously depends on the accuracy of the data fitting needed to produce the required interpolated and extrapolated data, which in turn depends nearly entirely on the modeling quality of the employed plate height expression. The template file29 therefore offers the user the possibility to visually and quantitatively (via the sum of errors) assess the fitting quality before selecting the plate height model for the actual KP number determination procedure. The KP number approach certainly is not new. Decades ago, it was, for example, already customary to compare packed bed and open-tubular systems by tabulating a series of (N,t) data couples lying on the Knox and Saleem limit lines.15,22 The KP numbers produced by the template file now cover the entire (N,t) space and not only the Knox and Saleem limit. It is believed that the KP numbers provide the single possible solution to rightfully map and compare the performance of systems belonging to different SSGs. They directly provide the most important kinetic information in the right dimensions and obviate the need to find Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

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Figure 6. (a) (N,E0) plot assuming ∆P ) 400 bar of the performance of some typical packed beds (PB-1 µm, 9; PB-2 µm, [; PB-3 µm, 2; PB-5 µm, /; UHPLC-1 µm,32 b) and silica monoliths (MS(50)-A,17 O; MS(50)-B,17 0; MS(50)-C,17 4; MS(50)-D,17 ]; MS-PTFE(B),16 s), taken from the literature. The packed bed data (PB-1-5 µm) were obtained using eq 2 with c via eq 12 and k′ ) 1 and with φ ) 700. The respective monoliths have following domain and pore sizes: MS(50)-A, ddom ) 10 µm, dpor ) 8 µm; MS(50)-B, ddom ) 4.2 µm, dpor ) 2.8 µm; MS(50)-C, ddom ) 3.3 µm, dpor ) 2.2 µm; MS(50)-D, ddom ) 3 µm, dpor ) 2 µm; MS-PTFE(B), ddom ) 3.81 µm, dpor ) 2.23. (b) (N,t0/N2) plot of the same silica monolith data (taken at ∆P ) 400 bar, same symbol code as above) and the dp ) 1 µm nonporous particle UHPLC data shown in Figure 4 for respectively ∆P ) 400 (b), 1000 (9), 2000 ([), and 4000 (2) bar.

a common characteristic size to properly reduce the plate heights and bed permeabilities. Of course not all the above-defined KP numbers need to be considered at the same time. Depending on the type of application, one would only have to focus on the most important one or two numbers of peculiar interest for his specific application. If one would, for example, be interested in finding the support type yielding the fastest N ) 50 000 plate separation, one would simply have to compare the t50k values. If one would be interested in selecting the best possible system to be used as the seconddimension column in a 2D LC system, where the available analysis time per peak in the second dimension would, for example, only be 10 s, it would suffice to compare the N10s and Neff,10s numbers of the different possible supports. If one would be interested to know how much peaks a given system can separate in a 2-h run, one could directly focus on comparing the np,2h numbers. To increase its general applicability, the posted template file29 allows the user to freely select the t and N values for which the KP numbers are reported. To rule out the influence of different employed sample and mobile-phase conditions, it would be recommended to report Neff and teff,N values, such that all supports would be compared on an 4066 Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

equal resolution basis. Since the values of Neff and tR, however, depend strongly on k′, it would remain, however, necessary to mention this k′ value each time a KP number is given. A means to circumvent this would be to systematically report the KP numbers for k′opt, i.e., the value of k′ yielding the lowest possible curve in an (Neff,Eeff) plot (cf. Figure 2b). Since most researchers automatically try to report the best possible performance of their system, this convention will probably be very easily adopted. Extrapolation of KP Numbers in the N < Nmin Range. The MS Excel template file29 also incorporates a function to make a fully safe extrapolation of the analysis times for plate numbers that are smaller than the leftmost point (Nmin) of the experimental (N,t0) data. This extrapolation is an important issue. Normally, it would be based on one of the existing plate height models, but since there is no real theoretical ground for any of these models, and since they can have a quite different extrapolation behavior in the small-N range (i.e., the large-u0 range), all model-based extrapolations should be used with the greatest caution. One possible approach to (partly) circumvent this problem would be that everybody would automatically select the model available in the template file yielding the most conservative estimate in the N < Nmin range. In addition, it is proposed to warn that these values

do not relate to truly measured data points by marking all KP numbers that are obtained in the N < Nmin range with a “*”sign (see Figure 5). A perfectly safe (i.e., underestimating) extrapolation can be obtained by making a linear extrapolation (boldface line in Figure 5) instead of using any of the model extrapolations. This linear extrapolation line is based on the assumption that the Nmin point corresponds to a velocity that is large enough to be purely C-termdominated, such that H = Clinu0. From eqs 6 and 7, it then follows immediately that

t0,lin ) ClinN

(18)

Since all models predict a linear increase of H with u in the purely C-term-dominated range, it can under the given assumption be expected that all experimental data points collected at a larger velocity than that corresponding to the Nmin point would all satisfy eq 18. The posted template file29 automatically draws the linear extrapolation line based on eq 18, after having determined the value of Clin from the coordinates of the Nmin point. This linear extrapolation line in fact represents the upper limit of all possible model extrapolation curves, because either the above assumption holds for the Nmin point, in which case all models coincide with eq 18 in the N < Nmin range, or else the Nmin point is still partly affected by some A-term band broadening, in which case the (N,t0) curves predicted by the model will descend more steeply with N than the linear decrease predicted by eq 18. It can easily be verified that the bottom dashed line in the pressure-varied Figure 4 corresponds to the linear extrapolation line defined by eq 18. This implies that eq 18 in fact represents the trajectory of the complete series of experimental (Nmin,t0) data points which would be obtained in experiments performed at a smaller pressure than the reference ∆P value used in eqs 6 and 7. The linear extrapolation line defined by eq 18 can hence be considered as a collection of actually measurable experimental data points. As a consequence, the “*” sign introduced to warn for the fact the cited KP number depends on a model-based extrapolation can be omitted. Figure 5 illustrates the definition and the use of the tN and tN* numbers in the N < Nmin range. The t0,1K value determined by the model extrapolation carries a “*” sign, whereas the t0,1K value determined from the linear extrapolation line does not carry this sign, as it relates to a data point that would have been actually measured if a smaller pressure would have been applied. The posted template file29 automatically adds a “*”-sign to KP numbers obtained by a model-based extrapolation outside the N range covered by the experimental data set. It goes without saying that the extrapolation of the Nt and EN numbers can be treated in a fully similar manner as the N numbers. It is also easy to unerstand that the difference between the KP numbers with and without a “*” sign in the N < Nmin range would automatically disappear if the Nmin point would be obtained at a velocity large enough to be purely C-term-dominated. It is hoped that the use of the “*” sign to stigmatize the model-extrapolated KP numbers in the template file29 will stimulate all researchers to measure the plate heights of their supports up to a sufficiently large mobile-phase velocity. For supports with a large A-term band broadening this might, however, turn out to be rather difficult,

as, for example, the requirement for high-speed detectors puts an upper limit to the range of applicable mobile-phase velocities. APPLICATION TO SOME TYPICAL LITERATURE DATA SETS To demonstrate their advantages, we applied the currently described kinetic plot and KP number methods to a number of typical packed bed and silica monolith plate height data taken from the literature. Whereas in a (u0,H) or (u0,E) plot no direct kinetic comparison can be made, kinetic plots such as the one in Figure 6a show at glance that packed bed columns and the current generation of silica monoliths each have their own range of N values where the one outperforms the other. Packed beds with particle sizes below or equal to 3 µm are clearly superior in the N < 50 000 range, whereas silica monoliths are to be preferred in the N >50 000 range. Figure 6a pinpoints the relatively poor performance of the silica monoliths in the N < 50 000 range to the fact that their Nopt values lie too far outside this range. From eq 16, we know that this message is equivalent to saying that the current generation of silica monoliths has too large pore and skeleton sizes to perform well in the N < 50 000 range. Obviously, the best performing silica monolith data ever reported in the literature is the MS-PTFE(B) silica rod.16 This monolith type beats all the other monoliths over a very broad range of 50 000 < N < 2 000 000. Below N < 25 000, it however yields t0 times, which are clearly larger than what can be expected for 2- and 3-µm particle packed beds. The reader should note that the impractically large Nopt values for most of the silica monoliths in Figure 6a do not mean that these monoliths cannot be used for small plate number separations. The large Nopt values simply follow from the assumption that ∆P ) 400 bar. For smaller reference pressures, for example, 4 bar instead of 400 bar, the obtained Nopt values would have been 100 times smaller and would hence have fallen in the more practically relevant range of N ) 104-105. This should, however, not be considered as an argument to redraw the (N,E0) plot at a smaller reference pressure, because, as can be noted from Figure 4, each decrease of the reference pressure inevitably leads to an increase of all t0 values, i.e., to a global underestimation of the kinetic performance of the tested support. One of the general conclusions that can be drawn from the kinetic plot shown in Figure 6a is that it currently seems to be impossible to decrease the minimal separation impedance of LC supports without also increasing Nopt. This can, for example, be noted by comparing the four capillary monolith data series. Going from the MS-50(D) to the MS 50(A) case, the E0,min values systematically decrease, but considering the corresponding domain and pore sizes given in the figure caption, these decreasing E0,min values have clearly been achieved by increasing the pore and domain size. As a consequence, the Nopt values shift toward impractically large N values (cf. the dref dependency of Nopt in eq 16). The shaded area added to Figure 6a indicates the region in the kinetic plot space where currently no supports are available. To intrude into this so-called “forbidden region”, novel monolith synthesis methods will have to be developed. Comparing the kinetic plots of the four considered capillary monoliths (series MS 50A-MS 50D) also makes it very clear that the increasing pore and domain size leading to the decreased separation impedance (E0,min) also leads to an increase of the Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

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Table 1. Values of the Typically Used Column Performance Characteristics for the Different Literature Data Series Shown in Figure 6a silica monolith MS(50)-A

MS(50)-B

MS(50)-C

MS(50)-D

MS-PTFE(B)

PB1 µm

PB2 µm

PB3 µm

PB5 µm

UHPLC1 µm

11.22 0.36 1.12 3.60 130.0 77 97

9.46 0.47 2.25 1.97 25.0 71 358

8.74 0.71 2.65 2.33 15.0 73 510

8.01 1.22 2.67 3.66 8.0 113 801

6.58 0.76 1.73 2.89 17.0 85 255

1.66 3.61 1.66 3.61 0.1 700 1936

3.33 1.80 1.66 3.61 0.6 700 1936

4.99 1.20 1.66 3.60 1.3 700 1935

8.31 0.72 1.66 3.61 3.6 700 1936

1.89 1.81 1.89 1.81 0.3 398 1418

Hmin (µm) uo,opt (mm/s) hmin ν0,opt Kv (× 10-14 m2) φ E0,min a

packed bed

See caption of Figure 6 for origin of data.

Table 2. Values of a Selection of KP Numbers Yielded by the Posted Template File29 for the Different Literature Data Series Shown in Figure 6a silica monolith

Nopt t0,opt (s) t0,1k (s) t0,25k (s)

packed bed

MS(50)-A

MS(50)-B

MS(50)-C

MS(50)-D

MS-PTFE(B)

PB1 µm

PB2 µm

PB3 µm

PB5 µm

UHPLC1µm

12 560 666 400 237 3.20* 88*

1 556 625 12 571 1.90* 62*

1 395 059 11 993 1.60* 57*

356 804 1 793 1.60* 54*

1 313 651 9 829 2.00* 56*

8 743 8 0.20* 159

34 971 122 0.55* 33

78 833 621 1.14* 51

218 562 4 771 2.90* 108

27 411 53 0.49* 23

a See caption of Figure 6 for origin of data. The “*” signs accompanying the t 0,1k and t0,25k numbers are automatically added by the posted template file to warn for the fact that these data relate to a model-based extrapolation outside the N range covered by the experimental data set after putting ∆P ) 400 bar.

analysis time in the N < 50 000 range. For most practical applications, the morphology of the MS(D) monolith, having the smallest pore and skeleton size, is hence to be preferred over that of the three other capillary monoliths and also over that of the MS-PEEK(B) silica rod monolith. This is a conclusion that cannot be drawn from any of the usually employed performance characteristics given in Table 1: based on the values for hmin, φ, Kv, or E0,min, the MS(A) column would be selected as the best performing silica monolith ever, whereas this in fact obviously is the worst choice if one is to do a separation requiring, for example, 25 000 plates. The KP numbers defined in the present study allow for a much more straightforward support selection. This can readily be noted from Table 2, providing a selection of the KP numbers yielded by the posted template file for the different data series shown in Figure 6a. Apart from the most important general KP numbers (Nopt, topt, t0,1k), Table 2 also contains the t0,25k number, which is of course the most direct and practically relevant KP number for the considered N ) 25 000 plate example. Based on the t0,25k number, the dp ) 2 µm packed bed immediately stands out as the best possible support system, without any need for further considerations or calculations. The other KP numbers given in Table 2 allow comparing the different support geometries in a more general sense. The Nopt number, for example, directly allows classifying the different support types according to their practical application field: either for easy separations requiring small N or for more difficult separations requiring large N. None of the numbers in Table 1 offers this possibility. The topt values show how well the given support performs around its Nopt value and can hence be used as a first indication of its overall kinetic performance. The t0,1k number on the other hand gives a very good indication of the kinetic 4068 Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

performance in the range of plate numbers running roughly up to Nopt/4 or Nopt/2. This can be appreciated from the fact that the classification of the five monolithic supports and the dp ) 3- and 5-µm packed beds based on either the t0,1k or the t0,25k numbers runs perfectly parallel. Another possible characteristic measure describing the kinetic performance in the N , Nopt range would have been the t0,1 time, i.e., the time needed to generate one theoretical plate (teff,1 in case of effective plate numbers), and being equal to the Clin constant defined in eq 18. The t0,1 point, however, usually falls too far outside the range of experimentally determinable (N,t0) data. As a consequence, it requires a much stronger extrapolation than the t0,1k number and is hence much more prone to extrapolation errors. Returning to Table 1, it can also be noted that assessing the quality of a given support type based on its E0,min value alone is not sufficient. In fact, the E0,min number can only be compared in a relevant way for supports with a comparable Nopt. For example, the MS-50(A) monolith has an Emin of 90, whereas the packed bed of 2-µm particles has an E0,min of ∼2000, but one would nevertheless clearly prefer the latter to perform a separation requiring N ) 25 000 plates (cf. the t0,25k values in Table 2). This is simply due to the fact that both systems reach their respective E0,min value in a totally different range of plate numbers, which in turn is a consequence of the fact that the MS-50(A) monolith has a much larger domain size than the dp ) 2-µm packed bed (cf. eq 16). More generally, it should be concluded that comparing separation impedances of different supports only makes sense if the value of N they are related to is mentioned simultaneously. This justifies the use of the EN numbers, because in this approach the cited E number is automatically related to a specific value of N. Another example showing that a comparison based on the E0,min

number alone is not sufficient comes from the two packed bed data series in Figure 1: whereas packed bed 1 has a significantly smaller E0,min than packed bed 2 (Emin ) 2351 versus Emin ) 2732 to be precise), the latter system yields shorter analysis times in the whole range of plate numbers below N ) 45 000. Kinetic plots are also perfectly suited to investigate the effect of an increased pressure drop, as currently pursued in the UHPLC approach. To demonstrate this, we removed the porous packed bed data from Figure 6a (for reasons of clarity) and replaced them by the nonporous dp ) 1 µm UHPLC data shown in Figure 4 to yield Figure 6b. To represent the influence of the increased pressure, the E0 number can no longer be used, and only the t0/ N2 ratio can be retained on the y axis. As can be noted, the effect of the increased pressure is 2-fold: it shifts the kinetic curves toward the left; i.e., larger Nopt values can be achieved, while simultaneously all t0 times decrease. The adopted kinetic representation mode hence clearly visualizes the advantage of the use of ultrahigh pressures as a means to invade the “forbidden region”. The presented UHPLC data should, however, be moderated by noting that they relate to capillary columns, where temperature gradient effects are small.33 CONCLUSIONS Traditionally, LC sytems are characterized by (u0,H) and (u0,E) plots. As an alternative we propose the use of kinetic plots. Only requiring the simple multiplications and divisions shown in eqs 6 and 7, these plots can be established very easily. Without the need for any numerical optimization routine or fitting procedure, they translate the experimental plate height data into the kinetic information one is finally after anyway, i.e., the minimal time needed to make a separation requiring N or Neff plates. Expressing the performance of a support in kinetic terms also circumvents the need for a common characteristic reference length to compare supports with a different morphology. Among the different possible kinetic performance plots, the (N,E) and (N,t/N2) plots offer the most detailed graphical comparison possibilities, whereas the (N,t) plots allow for a direct readout of the minimal analysis time. In the same effort, plots of the corresponding column length or the maximal peak capacity as a function of N can be readily established. Using a freely available MS Excel template file,29 requiring only the input of a series of experimental (u0,H) data, all possible kinetic plots can be automatically established. The file also contains a built-in function to automatically calculate a set of so-called KP numbers fully characterizing the obtained kinetic plot curves. If the performance of a given tested support is to be quantified with a minimal amount of information, this should hence preferentially be done by citing the data couple (Nopt, t0,1k) instead of the traditionally cited data couples (Hmin, Kv) or (hmin and φ). Given the current quest for speed, peak capacity, or both in LC, the former data couple is much more informative than the latter two. Similarly, the data triplets (Nopt,topt,t0,1k) are much more informative than the data triplets (Hmin,uopt,Kv). Obviously, these minimal KP number sets can be further extended with other KP numbers such as t10k, t50k, etc.. Compared to any of the other KP numbers, the Nopt number is somewhat ambiguous in the sense that it should not be as small or as large as possible, but simply needs to be as close as possible to the number of plates needed for a given application. If Nopt is larger than the required number of plates,

the feature sizes of the support need to be reduced and vice versa. To eliminate differences in the retention factor of the test components, which is often the case if supports with a small retention capacity such as highly open-porous monoliths or nonporous particles are involved, it is recommended (cf. the results shown in Figure 2) to communicate the performance of a given support by citing the Neff-based KP numbers (for example, Neff,opt, teff,opt, and teff,1k), preferably determined for the optimal retention factor k′opt. This would greatly help the standardization of column efficiency reporting. Applying the different kinetic plate height representation methods to a set of recent packed bed and silica monolith plate height data shows at glance that each range of desired plate numbers has its own optimal size and external porosity and that the current generation of packed bed and monoliths should more be considered as complementary rather than as competing systems. In the range of N < 50 000, packed beds with particles smaller than 3 µm are clearly superior to the current generation of silica monoliths, whereas in the range of N > 100 000, the silica monoliths yield unsurpassable performances. The method also clearly visualizes the potential advantage of the recently introduced UHPLC systems. ACKNOWLEDGMENT The authors greatly acknowledge a research grant (FWO KNO 81/00) from the Fund for Scientific Research-Flanders (Belgium). D.C. is supported through a specialization grant from the Instituut voor Wetenschap en Technologie (IWT) of the Flanders Region (Grant SB/1279/00). P.G. is supported through a specialization grant from the Instituut voor Wetenschap en Technologie (IWT) of the Flanders Region (Grant SB/13419). NOTE ADDED AFTER ASAP PUBLICATION This paper was posted inadvertently with errors in Figures 2 and 6a. The version posted on June 15, 2005 is correct. GLOSSARY a, b, c

constants in reduced Knox equation

A, B, C, D

constants in dimensional plate height equations (eqs 5 and 14)

Clin

C constant value based on the H value corresponding to the Nmin point of a given plate height series and defined as Clin ) H/u0 (s)

ddom

domain size (ddom ) dpor + ds) (m)

dp

particle diameter in packed bed (m)

dpor

flow-through pore size (m)

dref

reference length (m)

ds

skeleton size (m)

Dm

molecular diffusion coefficient (m2/s)

E

general separation impedance symbol, can be either E0, ER, or Eeff (/)

E0

t0-based separation impedance (E0 ) H2/Kv) (/)

Eeff

Neff-based separation impedance (Eeff ) ER(1 + k′)4/k′4) (/)

ER

tR-based separation impedance (ER ) E0(1 + k′)) (/)

E0,N, ER,N

t0- and tR-based separation impedance for a separation requiring N theoretical plates (/)

Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

4069

Eeff,N

tR-based separation impedance for a separation requiring N effective plates (/)

t0,opt, tR,opt

t0 and tR times needed to obtain Nopt theoretical plates (s)

h

reduced theoretical plate height (h ) H/dref) (/)

u0

H

height equivalent of a theoretical plate (m)

mean velocity of permeating, but nonretained solute (m/s)

k′

phase retention factor (/)

u0,opt

nonretained solute velocity yielding the minimal plate height Hmin (m/s)

(m2)

Kv

column permeability based on u0

L

column length (m)

Greek Symbols

n, m

exponents in Giddings and Knox equation (eqs 5 and 14) (/)

∆P

pressure drop (Pa)



external porosity (/)

φ

flow resistance (φ ) dref2/Kv) (/)

η

dynamic viscosity (kilo‚m-1‚s-1)

ν0

reduced fluid velocity (ν0 ) u0dref/Dm) (/)

np

peak capacity (/)

np,t

peak capacity which can be resolved isocratically in t seconds (/)

N

plate number (/)

Neff

effective plate number (Neff ) Nk′ 2/(1 + k′)2) (/)

Subscripts

Nmax

maximally achievable plate number, obtained by letting t f ∞ (/)

0

referring to t0

eff

referring to effective plate number

lowest plate number of an experimental plate height data set, found by filling in the largest considered u0 velocity into eq 6 (/)

Nmin

exp

experimental values

lin

referring to linear extrapolation

max

maximal

min

minimal

Nopt

plate number yielding the best possible analysis time/pressure cost ratio (see Figure 3) (/)

Nt

number of plates which can be generated in t seconds (/)

opt

optimal

R

tR-based quantity

Neff,t

number of effective plates which can be generated in t seconds (/)

ref

general reference measure

t

general time symbol, can be either t0 or tR (s)

t0

dead time of a column (s)

tR

retention time, related to t0 according to tR ) t0(1 + k′) (s)

tN

time needed to realize N plates, t can be either t0 or tR (s)

teff,N

retention time needed to realize N effective plates (s)

t0,N, tR,N

t0- and tR times needed to generate N theoretical plates (s)

4070

Analytical Chemistry, Vol. 77, No. 13, July 1, 2005

Other *

denoting KP numbers obtained using one of the plate height models but extrapolated outside the N range covered by the experimental data set

PB

packed bed

SSG

self-similar structures group

Received for review January 26, 2005. Accepted April 4, 2005. AC050160Z