Contrary to what could be expected for a technique whose

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1 pubs.acs.org/ac Downloaded via on December 20, 2018 at 06:41:45 (UTC). See for options on how to legitimately share published articles. Graphical Data Representation Methods To Assess the Quality of LC Columns We discuss the most important plot types for the kinetic performance of liquid chromatography columns and elaborate on how these plots should best be constructed and can be made dimensionless. Distinction is made between plots that are most suited for practitioners (column users) versus those most suited for theoreticians and column manufacturers. Gert Desmet,*, Deirdre Cabooter, and Ken Broeckhoven Vrije Universiteit Brussel, Department of Chemical Engineering, Pleinlaan 2, 1050 Brussels, Belgium KU Leuven University of Leuven, Department for Pharmaceutical and Pharmacological Sciences, Pharmaceutical Analysis, B-3000 Leuven, Belgium REPRESENTATION MODES MOST RELEVANT FOR COLUMN USERS Theoretical Background. Depending on the complexity of the sample, the main quality measures for a chromatographic separation are the resolution R s of the critical pair and the peak capacity n p. 11,16 21 In general, R s and n p depend on the efficiency N of the column and the phase retention factor k: Contrary to what could be expected for a technique whose foundations were laid in the 50s and 60s of the previous century, column and instrument technology for liquid chromatography has developed very rapidly in the past decade To represent the impact of these advances in a quantitative and comprehensive way, a great variety of different graphical representation modes exists. 2,4,11 15 The present feature article aims at providing an overview of these different plots and point out their best use. The paper is divided in two main parts: one on plots giving the best and most fair representation of the practical chromatographic performance (= information relevant for users) and one on plots giving insight in why a given performance is achieved (= information relevant for theoreticians and particle and column developers and producers). Emphasizing this distinction is important. Whereas the chromatographic performance is traditionally represented via plate height or van Deemter plots, this type of plot has originally been developed by theoreticians to study the different sources of band broadening. 2,11 The number of pure practitioners (column users) is however rapidly growing within the chromatographic society. They would be better served by directly getting a view on the efficiency or peak capacity that can be achieved (either in isocratic or gradient elution) and the time required to get it. It is therefore surprising to see that relatively little use is made of the possibility to directly plot efficiency versus time, although these measures are readily available in the analysis report issued by the instrument software. R n n s = N ( k2/ k1) 1 k2 4 k2/ k1 k2 + 1 (1) N = 1 + ln( k + 1) = N g 4 ( k ) p,iso last iso last p,grad = 1 + = 1 + Ngrad 1 b + 1 ln e bk 1 last b b b Ngrad g ( k ) grad last 4 wherein b is Snyder s gradient steepness factor, and N grad is the plate number defined by Neue. 16,17 The ratio k 2 /k 1 corresponds to the selectivity and k last is the phase retention factor of the last eluting compound (g iso and g grad are used in eq T-8 and are defined via eqs 2 and 3). k can be considered as the dimensionless time spent in the stationary phase (retained state) compared to the time spent in mobile phase (unretained) (k =(t R t 0 )/t 0 ). This definition holds irrespective of the elution mode and can hence be used for both isocratic and gradient separations without any restriction. The measured peak capacity n p can also be transformed into an observed efficiency N PC (see eq T-8 in Table 1). This quantity cannot be used for a direct theoretical analysis, because it is the convoluted result of a mix of efficiencies originating from multiple analytes (with different diffusion coefficients) and, in the case of gradient elution, also from a range of different mobile phase compositions. From a user s perspective, however, plots reporting this convoluted efficiency would be very valuable because N PC combines information from a multitude of analytes who furthermore have all experienced a Published: April 24, 2015 (2) (3) 2015 American Chemical Society 8593

2 Table 1. System and Performance Parameters Most Relevant for Column Users and for Column Producers wide range of different diffusion and retention coefficients and is hence much richer in information than the efficiency N which only relates to a single component and a single mobile phase condition. Table 1.2a provides an overview of some of the most widespread expressions to determine n p or R s from experimental chromatograms. Equations T-4 and T-5 reflect the different peak capacity definitions discussed in literature. 17,21,22 For the purpose of column characterization, it suffices to note that the qualitative result of the comparison will remain the same provided one consistently uses the same definition. Flow Rate- or Velocity-Based Plots. From a user s perspective, it would be most straightforward to look at plots of efficiency or peak capacity (or any of the quantities listed in Table 1.2a) versus any of the system variables F, ΔP, u (Table 1.1). Probably the most practical plots in this sense would be those involving the flow rate F. Figure 1 shows a number of possible combinations. As can be noted, the comparisons based on N, N PC, or n p give very similar results (note that peak capacity is only proportional to the square root of N). This is in agreement with theory, because the data were measured at constant k. 23 For this particular example, column FP produces a significantly better separation quality than column CS (which has a shorter column length and N scales with length). A slightly different view is obtained for the resolution (Figure 1d). This should however not be considered as a deviation, because R s is a different measure, strongly reflecting the selectivity between two specific compounds. It is hence much more component-specific than N or n p. Plots of R s versus F or ΔP can be very useful during the ultimate optimization step in method 8594 Figure 1. Examples of system-parameter based efficiency plots comparing the performance of columns packed with core-shell (CS = Kinetex C18 (100 Å) 75 mm 4.6 mm, 2.6 μm) and fully porous (FP = ACE C18 (100 Å) 150 mm 4.6 mm, 3 μm) particles. Test conditions: (a,d) isocratic elution, k = 7; (b,c) gradient elution, k =2 12. Data reformatted from results of an experimental study (see ref for experimental conditions). 23 development, e.g., to indicate the range of flow rates where R s 1.5. Fixed-Length Kinetic Plots (N and t Achievable on a Column with Given Length). Whereas the plots in the previous section provide a clear and direct view on the

3 achievable separation quality, they lack information on the corresponding analysis time. The latter is important because, when mobile and stationary phase have been optimized, resolution or peak capacity can only be increased at the expense of time. The optimization of resolution and peak capacity should therefore preferably go hand in hand with the optimization of the analysis time. 11,24,25 Plots of separation quality versus time (or vice versa), generally referred to as kinetic plots, provide the ideal tool to study this combined optimization. 15 Establishing such plots is in fact very straightforward, as time is a parameter which can be directly read-out from the chromatogram. 26,27 The times most relevant for the user are the retention time t R of the last eluting compound (because this determines the duration of the analysis) and the t 0 -time (because this is the basis for the selectivity and the elution window). Figure 2 shows two examples of such fixed-length Figure 3. Examples of variable-length kinetic plots comparing the kinetic performance limit (KPL) of (a,c) two core-shell columns with different particle sizes and (b,d) four theoretical data sets A, B, C, and D. Data in parts a and c are rescaled and reformatted from experimental results obtained on a core-shell HALO C μm column; 23 see Figure 6 for properties columns A, B, C, and D. ΔP max = 1200 bar. Figure 2. Examples of fixed-length kinetic plots comparing the performance of the same columns as shown in Figure 1. kinetic plots, reconsidering the data from Figure 1 but now putting the measured time on the x-axis instead of F or u 0. The plots readily show that column FP indeed provides a higher separation quality than column CS but achieves this at the expense of an increased analysis time. Figure 2 in fact readily shows that both columns offer complementary efficiencies in their C-term region, i.e., the region left to the curve maxima. In general, many other fixed-length kinetic plot variants can be conceived of plotting any of the experimentally determinable quantities in Table 1.2a versus any of the times in Table 1.2c. Kinetic Plots with Variable-Length (Kinetic Performance Limit-Plot, Pressure = ΔP max ). A drawback of fixedlength kinetic plots is that the information depends on the column length, with larger lengths generally producing a larger separation quality (cf. the dotted lines for increasing lengths shown in Figure 3a). In a series of papers, we have shown how the separation quality data measured on a single column with given length can be directly transformed into a curve enveloping all possible fixed column length curves. 15,23,26 31 This enveloping curve (cf. the full lines in Figure 3a) is generally referred to as the variable-length kinetic plot and represents the kinetic performance limit (KPL) of the investigated material, for there exists no combination of column length and flow rate with which the tested material would yield a better performance. Table 1.3 provides the simple expressions allowing to calculate the KPL-curve from a series of measured times and separation qualities on a column with fixed-length (subscript meas in the table). Two different but equivalent approaches can be used to calculate the KPL-data: the λ-method and the plate height method. 27,23 Figure 3 shows some examples of variable-length kinetic plots. Many other examples can be found in the literature. 25,32 45 Figure 3a,c displays the same data set, relating to two different sizes of a core-shell particle. Both plots clearly emphasize the complementary of both particle sizes, with the smaller particles performing best for short separations (small t 0 ), while the opposite holds in the range of separations requiring large times. The type of plot shown in Figure 3c is traditionally referred to as the Poppe-plot. 13,46 Figure 3b compares four hypothetical columns A, B, C, and D. The same data are replotted as a zoomed kinetic plot (Table 2.3) in Figure 3d, where t on the y-axis is divided by N 2 to magnify the differences in time for the same N. Plotting the axis as 1/N instead of N, the zoomed plot furthermore reflects the shape of the underlying van Deemter curve, readily showing the B- and C-term dominated range of velocities, respectively, on the left and right-hand side of the curve minimum. 15 The advantage of variable-length over fixed-length kinetic plots is that they directly show the full optimized performance for any possible column length. Information about the column length with which a given optimized performance can be achieved can be readily calculated (or plotted) by using the expressions for L KPL given in eq T-15. Fitting Functions and Key Equations. Any good method to graphically represent experimental data requires that these data can be fitted to a theoretical model using a simple expression. The appropriate theoretical basis for any of the different chromatographic separation quality parameters is the plate height model, of which many different variants have been proposed. 47,48 Some of the most frequently used plate height models can generally be written as n B H = Au0 + + Cu0 u (4) 0 which for n = 0 leads to the van Deemter-variant (A-term is constant), while the n = 1/3-case represents the so-called Knoxvariant. 2,

4 Table 2. Overview of the Most Relevant Plot Types To establish variable-length kinetic plots, the required key equations are given by eqs T-14, T-15, and T-18, which can be directly derived by expressing the link between column length, time, and velocity under conditions where the pressure drop is the limiting factor, i.e., kept at its maximum value ΔP max : L = NH = ut (5) u = K ΔP /( ηl) v max (6) where η is the viscosity of the mobile phase and K v is the column permeability, based on either the interstitial or linear velocity (see Table 1). From these expressions, the relation between N and t 0 can be obtained from N = L/H and using eqs 5 and 6 to replace u 0 in eq 4: 50 tkpl NKPL( tkpl) = n α t + β t + γ (variable length kinetic plot) KPL KPL (1 )/2 KPL KPL KPL (7) wherein α, β, and γ are three fitting constants, which can, respectively, be retraced to the A, B, and C-constants in eq 4. To establish fixed-length kinetic plots, the key equations are even simpler since now L is fixed. Hence, eq 5 can be readily used to calculate H and t. Furthermore, the velocity u 0 appearing in eq 4 can readily be replaced by the ratio L/t 0. This leads to an expression of the form: N ( t ) t = αt + βt + γ (fixed length kinetic plot) (1 n) 2 (8) Working at constant k (see section Selection of Sample and Test Conditions), these equations can also be used to directly fit peak capacity or resolution-plots by using them to replace N by t in the basic expressions for n p or R s (eqs 1 3). 51 Replacing t and N by, respectively, t and N (eqs T-27 and T-28), eqs 7 and 8 can also readily be used to fit dimensionless kinetic plot curves (see further on). Selection of Sample and Test Conditions. The analyte(s) used to determine the separation should be selected with care. 34,23 First, this depends on the specific purpose of the planned comparison. Aiming at a generic column characterization, a mix of compounds eluting over a wide retention range without exhibiting any specific interactions leading to peak tailing or fronting should be preferred (e.g., mixtures of structural homologues such as alkylbenzenes or alkylphenones). Alternatively, when trying to select the best chromatographic medium for a specific separation problem, the experiments should evidently be done with the sample of interest or at least with a subset of its most critical components. In many cases, columns are characterized using a single component but, as already mentioned, this is mostly useful for pure theoretical work or to understand column performance. To generate plots for practitioners, it would be more interesting to report a convoluted measure such as n p or N PC, preferably based on a generic test mixture representing the different classes of components typically analyzed in the lab. Such samples can, for example, contain components with a broad range in log D (i.e., polarity or hydrophobicity), such that the resolution between early eluting compounds can be studied together with the overall peak capacity. For gradient elution, separation resolution, peak capacity, etc. depend strongly on the effective retention factor k. To obtain the fairest comparison of columns packed with the same stationary phase but used at different F, or having a different column length L, it is hence important that the compounds of interest (critical pair or last compound) elute at the same k. This is achieved by changing the gradient time t G in proportion with the change in t 0 -time, so as to keep the same t G /t 0. If the dwell time t dwell is significant, the ratio of t dwell /t 0 should be maintained constant as well, e.g., by introducing an additional isocratic hold at the start of the gradient for longer or broader columns. 27,23,29 When comparing columns with different retention properties, any attempt to elute one or more key components at the same or at least a similar k usually requires switching to a (slightly) different mobile phase or gradient program. This inevitably implies that the comparison is made for a different viscosity and molecular diffusion coefficient. If desired, this bias can be eliminated by moving to a dimensionless representation of the kinetic performance (see further on, Table 2.5). Alternatively, the influence of viscosity and molecular diffusion could be left as it is. From a user s perspective, this would in 8596

5 fact be the preferred option, for it would reward columns that can do the analysis with a lower mobile phase viscosity or a higher molecular diffusion coefficient. A nice example for this is the kinetic advantage of HILIC versus RPLC separations recently made by Heaton et al., where the kinetic performance of the same compound (nortriptyline) was compared in RPLC and HILIC mode. 52 Figure 4 in this reference shows a decrease Figure 4. Difference between plots based on u 0 and t 0 (dotted lines) and plots based on u i or t i (full lines) for 1.7 μm fully porous particles (FP 1 ) and 2.6 μm core-shell particles (CS 1 ). (a) Plate height plot of H versus u and (b) kinetic plot of t i versus N PC with ΔP max = 1200 bar. Column CS 1 : ε int = 0.20 (ρ = 0.65). Column FP 1 : ε int = 0.40 (ρ = 0.65). In each case: D pz =10 10 m 2 /s, ε = 0.40, k =5,D mol =10 9 m 2 / s. of roughly 40% in analysis time in HILIC mode to obtain the same efficiency and an improvement in efficiency of around 30% for the same analysis time, in agreement with the 40% lower viscosity of the mobile phase. 24 REPRESENTATION MODES MOST RELEVANT FOR COLUMN PRODUCERS Whereas users are more concerned about the actual separation quality they can get in a given time, column producers are also interested in finding out why this performance is obtained and where the margins for improvement lie. The best way to identify these is by using plots where the different factors influencing the performance are eliminated, either one by one or in groups. Typically, this elimination leads to working with dimensionless variables, such as the reduced plate height h, the reduced velocity ν. Theoretical Background. Since N, R s, and n p strongly depend on the column length L, this is the first obvious system parameter that needs to be eliminated. For the separation efficiency N, this gives rise to the plate height concept (H = L/ N). Adopting the general plate height model, the plate height H can be linked to the average interstitial velocity u i, the main packing parameters (ε = external porosity, d p = characteristic size solid zone, α = contact area between stationary and mobile zone divided by stationary zone volume, mass transfer shape factors Sh m and Sh pz ) and the physicochemical parameters of the analyte and the mobile and stationary zone (zone retention factor k, D mol, D pz ). 3,12,53 58 The general plate height model usually splits up the band broadening in four main contributions: the eddy-dispersion (H eddy ), the longitudinal diffusion (H B ), the mobile (H Cm ), and the stationary zone (H Cs ) mass transfer resistance (the latter two occur on the particle scale). 59 From their physical meaning, it makes most sense to relate the H Cs -contribution to the characteristic size of the solid zone (d p ), the H Cm -contribution to the size of the through-pores (d por ) and the eddy-dispersion (representing the effect of packing disorder on the scale of multiple flow-through 8597 paths) on the so-called domain size (d dom ), which, to a first approximation, can be taken as the sum of d p and d por. 56,60 D H = H d + + k + C Sh α k u d eff eddy( dom) 2 (1 ) m( m,, ) i u D i p 2 por 2 + C Sh α k u d s( s,, ) i Dpz (9) In a general chromatographic medium, d p and d por are independent of each other and are difficult to relate in a general way. In a packed bed of spheres (α = 6), the link between both sizes is traditionally made via the hydraulic equivalent cylindrical through-pore size (d por = (2/3)d part ε/(1 ε)). 61 In this case, the expression for H can be written as (d p = d part ): 55,56,62 Deff H = Heddy( ddom) + 2 (1 + k ) u 1 k ε (1 + k ) 1 ε 2 i ud i Sh D m part 2 mol 1 k + 3 (1 + k ) 2 mol ud i Sh D part 2 part pz (10) Note that the fundamental retention parameter in eqs 9 and 10 is not the phase retention factor k but the so-called zone retention factor k, defined via the interstitial residence time t i (see eq T-13). 63,64 In the subsequent sections, all column data were produced using eq 10. This model-based approach offers the advantage that the eddy-dispersion can be kept identical (realized here by always taking Giddings coupling theory expression h eddy = (1/A + 1/(Dν i )) 1 with A = 1.2 and D = 0.7) while changing any of the other packing parameters (intraparticle diffusion, presence of core, intraparticle porosity, external porosity). This allows to illustrate in a simple manner which plots reveal the purest possible information about the packing quality (= structural homogeneity). Eliminating the Effect of Length. Plots of H versus the velocity or the flow rate, generally referred to as plate height or van Deemter plots, are undoubtedly the most popular graphical representation of chromatographic performance. 2,11 An example is shown in Figure 4a. The plot is based on the (theoretically justified) assumption that the plate height is independent of the column length. 23,27,65,66 Under the same assumptions, plots of the length-reduced resolution or peak capacity can be made as well, e.g., by putting R s 2 /L or (n p 1)/ L on the y-axis. Plots of H PC = L/N PC (see eq T-8) can be made as well. 67 Eliminating the Effect of Particle Void Volume on Velocity and Permeability. The vast majority of van Deemter plots in literature use the mobile phase velocity u 0. This is also the case for the dashed curves in Figure 4a. The full curves on the other hand were plotted versus the mobile zone velocity u i, the average velocity of the part of the mobile phase that is actually in motion, i.e., the part outside the particles. 63,68 As can be noted, the slope of the C-term dominated regime of the plate height curve of the fully porous particles (FP 1.7 ) appears considerably less steep if plotted versus u i instead of u 0. This effect is much smaller for the core-shell particles (CS 2.6 ). These have a considerably smaller intraparticle void volume than the fully porous particles. Because of their large particle void volume, the latter have a larger part of stagnant mobile phase. Hence, in order to achieve the same u 0 -velocity, the velocity in the interstitial space needs to be much higher for the

6 fully porous than for the core-shell particles. Since H depends on u i and not on u 0 (eq 10) this inevitably leads to a higher observed C-term in a plot based on u 0. Column manufacturers or theoreticians wanting to compare packing quality or homogeneity via a plate height plot should hence rather base this on the u i -velocity instead of the u 0 -velocity, as this minimizes the interference from the intraparticle properties (see eq T-12 to see how u i should be calculated from the more easily measurable u 0 = L/t 0 ). For the same reasons, comparing packing quality between particles with different intraparticle void volume from the observed kinetic performance should preferably occur via kinetic plots based on the u i -based residence time t i (eq T-12) and separation impedance E i (= H 2 /K vi, see eq T-10 and T-21). Figure 4b compares the kinetic plots corresponding to the data set already considered in Figure 4a. As can be noted, the position of the fully porous particle curve (FP 1.7 ) considerably shifts toward the more advantageous the bottom right corner if plotted versus t i (full lines) than if plotted versus t 0 (dashed lines). This is again due to the fact that the t i -based representation is no longer influenced by the significant intraparticle contribution to the t 0 -time. Plotted versus t i, the FP 1.7 -curve crosses the CS 2.6 -curve, indicating a similar packing quality, in agreement with the fact that both data sets have been produced assuming the same h eddy -contribution (curves do not overlap because they relate to particles with different size). Eliminating the Effect of Size and D mol and Viscosity. To eliminate the effect of the particle or, more general, the solid zone size, two cases can be considered: one wherein the particle (or solid zone) size is known and one where it is not. Size Known. When the solid support size d is known, one can readily rule out its effect by switching from absolute to dimensionless plate heights (see Table 2.4). As far as we can retrace, the use of h = H/d p to eliminate the effect of size on H has been proposed by van Deemter et al. in They also proposed to eliminate the effect of the particle size on the velocity, by multiplying both quantities. This indeed works, in agreement with theory, but does not yield a dimensionless quantity yet. To achieve the latter, the product u d needs to be divided by the molecular diffusion coefficient of the analyte. 11,49,69 This yields the so-called reduced velocity ν, in chemical engineering literature usually referred to as the Pecletnumber. This parameter eliminates both the effect of size and molecular diffusion. 11 Applying this transformation to the u i - based curves from Figure 4a yields the h versus ν i -curves shown in Figure 5a. With the size effect being eliminated, column CS 2.6 (open symbols) now clearly performs better than column FP 1.7. Fixed and free length kinetic plots can be made dimensionless as well, introducing a dimensionless time t Figure 5. Reduced (dimensionless) variants of the plots shown in Figure 4a,b and a dimensionless plate number N (eqs T-27 and T-28). These expressions have been established by replacing H, u, and K v in the basic expressions for N and t by their dimensionless counterparts h, ν, and ϕ and by regrouping all dimensionless parameters on one side. Figure 5b retakes the two t i -curves of Figure 4b but now in dimensionless units. Whereas in Figure 4b both curves crossed each other because of the different particle sizes, the dimensionless CS 2.6 -curve consistently lies below the FP 1.7 -curve in Figure 5b, reflecting the better kinetic performance of the CS 2.6 -particles and hence reflecting the more advantageous model parameter values with which the CS 2.6 -data were generated. Size Unknown. When the size is unknown, for whatever reason, the closest estimate for the true size can be obtained by measuring the bed permeability and multiplying it with a theoretical expression for the expected flow resistance based on the measured external porosity ε: dϕ = K v ϕtheo (11) The subscript ϕ (flow resistance) is used to emphasize that d ϕ only represents an estimate for the characteristic size, obtained via the bed permeability. Either the u 0 -oru i -based permeability can be used (see eq T-22), provided they are multiplied with the corresponding ϕ theo -expression (respectively, ϕ theo,0 and ϕ theo,i, see eqs T-23 T-25). To calculate ϕ theo, the external porosity ε needs to be determined via any of the methods proposed in the literature. 70,71 For packed beds of spheres, ϕ theo is given by eq T-23, derived from the Kozeny Carman law. 72 For monolithic beds, where the through-pores usually have a much larger relative size, more specific correlations should be used (see example in eq T-24). 73,74 Once d ϕ is known, it can readily be used in any of the dimensionless numbers in Tables 2.4 and 2.5 as a replacement for d. This is illustrated in Figure 6, reconsidering the data sets already shown in Figure 3b,d, but now in a dimensionless format. Figure 6a shows the data in a flow resistance-reduced plate height plot (h ϕ versus ν i, ϕ, see Table 2.4). Whereas the representation as t 0 /N 2 versus 1/N in Figure 3d suggests a similar column performance for columns C and B (both have similar value at curve minimum in Figure 3d), this is no longer the case in the plot in Figure 6a because this format maximally eliminates the effects of permeability and external porosity. As can be noted, the data sets now nicely cluster up in two groups, in line with the underlying values of the band broadening model parameters on which they are based: one group for the two core-shell columns (columns A,C) and one for the fully porous columns (B,D). This type of plot hence reflects the pure band broadening properties of the columns in a much better way than the more user-oriented plots of t 0 /N 2 or t 0 versus N or 1/N plot shown in Figure 3b,d. Halasz proposed establishing dimensionless plate heights and velocities using the square root of the column permeability as the characteristic size (eq T-26). 14,75 Whereas these measures are also dimensionless, Figure 6b shows that this type of plot is still heavily influenced by the differences in external porosity ε (columns A and B have a higher ε than columns C and D), a problem which was also already obscuring the conclusions on the packing quality which could be made from Figure 3d. Yet another possibility to produce a dimensionless kinetic plot starting from Figure 3d is by replacing t/n 2 by the impedance E (= t/n 2 ΔP/η) and N by N opt /N. 15,43,76 An

7 Figure 7. (a) Examples of different packing geometries with increasing disorder. 83 (b) Plots of E ε versus 1/N ε for all data sets shown in Billen et al.; ordered from most homogeneous (1.5 μm-case) to most heterogeneous packing (12 μm-case). 83 Reprinted from ref 83. Copyright 2006 American Chemical Society. Figure 6. Examples of dimensionless plots comparing the performance of columns A, B, C, and D when the size of the packing is unknown. (a) Flow resistance-reduced plate height plot, (b) Halasz plot, (c) reduced impedance plot, 76 (d) ε-filtered impedance E ε versus 1/N ε. Column A (ρ = 0.65): ε = 0.40, D pz = m 2 /s, ε int = Column B (ρ = 0): ε = 0.40, D pz = m 2 /s, ε int = Column C (ρ = 0.65): ε = 0.35, D pz = m 2 /s, ε int = Column D (ρ = 0): ε = 0.35, D pz = m 2 /s, ε int = In each case: k =5,D mol =10 9 m 2 /s. advantage of this approach is that both the E- and the N opt /Nnumber can be calculated without having to specify a size (N opt is simply the N-value corresponding to the minimum of the zoomed kinetic plot, see, e.g., Figure 3d). A drawback of N opt is that it depends rather heavily on the B-term contribution and the permeability and therefore provides a less clean measure for the actual band broadening process. In Figure 6c, this is reflected by the fact that the clear clustering between the coreshell and the fully porous group visible in Figure 6a cannot be obtained with this type of plot. On the other hand, introducing the ε-corrected impedance E ε and reduced plate number N ε, see Figure 6d, (Table 2.6), the kinetic data which originally appear rather scattered in the user oriented-plots given in Figure 3b,d are in Figure 6d again clearly clustered following the same pattern as in Figure 6a, i.e., reflecting the underlying values of the band broadening model parameters with which the data were produced. Figure 7b shows a set of kinetic performance data measured (via numerical simulations) in a series of packing geometries with increasing disorder (Figure 7a). As can be noted, plotting the data in the external porosity-corrected format of E ε versus 1/N ε automatically arranges the curves from the most ordered (lowest curve) to the least ordered case (highest curves), showing that this type of plot is indeed excellently suited to identify differences in packing heterogeneity. Eliminating the Packing Shape Effect: Monolith vs Packed Bed. Because the different terms in eq 9 depend on different characteristic distances and because the mutual relation between these distances changes when the shape of the packing changes, it is theoretically impossible to define a unique geometrical measure that would allow one to eliminate the effect of this shape via a dimensionless plot. Considering that the external porosity ε is one of the main shape factors of a structure, this also implies that there exists no characteristic distance for which two monolithic columns with the same degree of packing heterogeneity but with a different ε would yield coinciding reduced plate height curves. 77 Only approximate measures, such as the domain size (d dom = d por + d p ) can be conceived of. 60 However, these have no sound theoretical basis and will inevitably either under- or overestimate the effect of ε depending on the test conditions. Stepwise Elimination of B-, C s -, and C m -Term Contributions. The most advanced analysis of packing quality can be made by subtracting (one by one) the known (or estimated) contributions of the most important individual band broadening sources from the total plate height value. 9,68,78,79 To visualize how this provides insight in the pure packing quality, Figure 8 represents four different sets of column performance, again referred to as columns A, B, C, D and again all calculated assuming the same packing quality (same h eddy ) but with slightly different intraparticle parameters (see captions). Figure 8a shows the complex mutual relation Figure 8. Example of subsequent subtracted reduced plate height analysis showing (a) h vs ν i, (b) h h B versus ν i, (c) h h B h Cs versus ν i, (d) h h B h Cs h Cm versus ν i. Column A (ρ = 0.65): D pz = m 2 /s, ε int = Column B (ρ = 0): D pz = m 2 /s, ε int = Column C (ρ = 0.65): D pz = m 2 /s, ε int = Column D (ρ = 0): D pz = m 2 /s, ε int = In each case: ε = 0.40, k =5,D mol =10 9 m 2 /s. 8599

8 Table 3. Procedures to Calculate H B, H Cs, and H Cm between the reduced total plate height curves of the four materials. The obvious first step to better understand this pattern consists of subtracting the B-term. 68,78 As shown in Table 3.1, this contribution can be determined by either measuring the plate height at very low flow rates or via the socalled peak parking method, invented during the famous Giddings and Knox taxi ride. 49,80 In principle both approaches yield similar results. 52 The impact of this subtraction can be assessed by comparing parts a and b of Figure 8. The second contribution for which a theoretically sound expression exists is that of the mass transfer resistance in the particles (H Cs -contribution). This can be estimated using the expressions in Table 3.2, accounting for the possible presence of a solid core. The most important parameter in eq T-34 is D pz, the effective diffusion coefficient in the porous zone (= entire particle for fully porous particle or shell-layer for coreshell particles). The expressions needed to estimate D pz are listed in Table 3.3. If a higher precision is required, the β 1 - expression in eq T-38 can be replaced by putting ζ = in eq 53a from Desmet and Deridder. 81 The impact of the H Cs -correction can be assessed from Figure 8c. The different curves for h h B h Cs now split up in two groups, dividing the two fully porous and the two coreshell particles and reflecting the more advantageous mobile zone mass transfer properties (lower C m because of lower k )of the core-shell particles. Possibly, one could also try to subtract the C m -contribution (3rd term in eq 10). However, some caution is required, because the only available h Cm -expression (eq T-41) has been established under the assumption that the flow in the interstitial space has a pure plug flow profile. Although this is the best approximation that currently exists, this assumption actually is far from correct. 59 In addition, the value of Sh m is ill-known. 62 Traditionally eq T-42 is used to estimate Sh m, but this correlation has been established for very high reduced velocities, lying outside the typical range encountered in HPLC. As has been shown, it is more correct to include a diffusion-based term in the expression for Sh m, cf. eq T In the theoretical example in Figure 8, Sh m is exactly known, such that the overlap of the curves for the different columns is perfect after the h Cm -correction (Figure 8d), hence properly 8600 reflecting that all columns were calculated by assuming the same h eddy -contributions. Eliminating the Effect of Retention Factor. The theory in section Theoretical Background shows that a fair comparison is only possible if running experiments at the same k. No corrective plot or data representation methods exist to fully eliminate the effect of k. The only partial correction for the k - effect can be made via the subtractive plots discussed in the previous section. CONCLUSIONS AND RECOMMENDATIONS Many different plot types exist to compare and analyze chromatographic performance (Table 2). Whereas the traditional way of representing chromatographic performance via (reduced) plate height plots should remain the dominant method for column development purposes, there is no good reason why column users should not get their information in a more direct way, i.e., via plots of the time needed to achieve a given efficiency, resolution, or peak capacity. The most simple of these would be a plot of t vs N (or vice versa) or, in dimensionless units (to remove the effect of particle size) t vs N (or vice versa). Plots Most Informative for Column Users. These plots should preferably represent the separation quality measure such as n p, R s, N PC,orH PC derived from a broadly composed sample or from a specific set of test compounds. Given the separation selectivity and the width of the achievable resolution window are determined by the phase retention factor k, plots for column users should be established by closely maintaining the same k for the key test components. Since k is thermodynamically related to the t 0 -time, the relevant velocity for user-based plots is the u 0 -velocity, with corresponding column permeability K v0. Combining the information on separation quality and time leads to so-called kinetic plots. These can either be made by plotting the crude data collected on the tested column (fixedlength, subscript meas in Table 1.3, ΔP = variable) or by rescaling the data to the maximal column length (subscript KPL in Table 1.3, ΔP = fixed = ΔP max ). Whereas the first type is specific for one column length, the latter provides a comprehensive view on all optimal combinations of separation quality and analysis time that can be expected from the tested

9 material. Zoomed kinetic plots (e.g., Figures 3d and 6d) offer a magnified view on the time differences for the same N. Plots Most Informative for Theoreticians and Column Producers. To steer their development process, column producers need to know how the performance observed in a user-oriented plot is influenced by the intraparticle properties on the one hand and by the packing structure on the other hand. This requires a more detailed analysis, involving plots where the intraparticle properties are maximally eliminated by basing them on u i or t i and using the u i -based permeability K vi instead of their more customary used t 0 -based counterparts. Experiments should rather be carried out at constant zone retention factor k, instead of at constant k (see eq T-13 for the difference between k and k). To rule out the effect of the particle size, dimensionless plate height and dimensionless kinetic plots can be made (involving the newly introduced t and N in eqs T-27 and T-28). When the characteristic size d of the packing is not known or difficult to define, d can be replaced by the hydrodynamic size d ϕ (eq T-22). To remove the effect of the external porosity ε from a zoomed kinetic plot, the ε-corrected impedance E ε and dimensionless plate number N ε can be introduced (Table 2.6). AUTHOR INFORMATION Corresponding Author * gedesmet@vub.ac.be. Notes The authors declare no competing financial interest. Biographies Gert Desmet heads the department of Chemical Engineering at the Vrije Universiteit Brussel, Belgium, where he teaches courses on biochemical and chemical reactor engineering as well as on analytical separation sciences. His research mainly focuses on the miniaturization and automation of separation methods as well as on the investigation and the modeling of flow effects in chromatographic systems. Deirdre Cabooter is assistant professor at the Faculty of Pharmaceutical Sciences at the University of Leuven (KU Leuven) in Belgium. 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