A series of theoretical, visualization, and simulation tools that are used to improve the structure and chemistry of the next generation of liquid chromatography (LC) columns is briefly reviewed. The article describes how this combination of visualization and simulation techniques can accurately predict column performance and support method development in a “dry-lab” approach. The potential for the next generation of chromatographic systems and columns to overcome the current resolution and speed limitations of one-dimensional chromatography is also explored. Potential developments include the next generation of narrow-bore columns packed with sub-1-µm particles for “ultra-ultra-high-pressure” liquid chromatography (UUHPLC) up to 7 kbar, high-throughput three-dimensional (3D)-printed technologies for delivering materials with feature sizes smaller than 5 µm, and the next generation of zero extracolumn volume systems for ultrafast gradient LC using 1-cm-long narrow-bore columns for sub-5-seconds routine gradients.

	High performance liquid chromatography (HPLC) is more than 50 years old (1) and has established itself as one of the most widely used techniques for separation and analysis. Over half a century, both systems and chromatographic columns have evolved dramatically. This development has mainly been driven by the insatiable search for faster analyses and more efficient columns, leading to the development of smaller fully porous particles (FPPs) (2), sub-3-μm core-shell particles (3,4), and to the application of ultrahigh pressures around 1 kbar (5). In 2020, ultrahigh-pressure (UHP) and ultralow dispersive LC instruments, which were introduced to the separation field 15 years ago to run columns packed with sub-2-μm particles, have not evolved much. They still remain the state-of-the-art technology that sets the limits of the speed-resolution power in unidimensional LC (6). As well as these instrument and column evolutions, the nature of the applications has also changed considerably over the past 50 years. LC now needs to adapt to new separation challenges from growing research areas in biology. From amino acids, peptides, proteins, DNA, monoclonal antibodies (mAbs), virusâlike particles to exosomes, new designs rivalling the conventional porous or superficially porous chromatographic materials are also needed (7).

	While trial-and-error, design of experiments (DOEs), and, more recently, artificial intelligence (AI) are frequently used to develop new structures and surface chemistries for targeted purposes, the underlying rules of physical chemistry still have a huge role to play in the design of new technologies in LC. To reduce research and development times in the industry, quality by design (QbD) approaches are preferred to random or statistical methods. However, they require expertise and strong fundamental understanding of the process. Given the complexity of the separation mechanism of complex mixtures, which involves knowledge in both adsorption (thermodynamics) and mass transport (kinetics) in a multiscale system, there is a constant push in the separation community towards bridging the persistent gap between the performance predicted by either statistical models or approximate theories and that observed by the chromatographer in the laboratory.

	The first goal of this article is to assess the power of well-known basic theories of chromatography (plate, rate, and stochastic theories) for the development of new chromatographic materials and to pinpoint their limits with respect to the actual phenomena of retention and band broadening taking place in real chromatographic columns. To that end, the advances in the physical reconstruction of the multiscale structure of an entire chromatographic column (macroporous and mesoporous space) combined with fluid dynamics, Brownian, and molecular dynamics simulation techniques (for the determination of actual flow velocity profile, diffusivity, and distribution of the analyte [8]) is discussed and illustrated in the case of size-exclusion chromatography (SEC). The second goal of this work is to report data on the latest progress in the development of “ultra-ultra-high-pressure liquid chromatography” (UUHPLC) using 3-cm-long narrow-bore columns packed with sub-1-μm particles (

 = 700 nm), three-dimensional (3D)-printed columns based on different technologies, and in ultrafast gradient LC using very short 1-cm-long columns. The potential of these three different techniques in the years to come will be discussed and assessed.

	For the isocratic runs, the flow rate is set at 0.45 mL/min, the volume fraction of acetonitrile in water is 65%, and the temperature was fixed at 27 ºC. The injection volumes were set at 1 μL and 10 nL when using an Acquity I-Class UHPLC system (Waters) and the reduced volume research prototype system, respectively.

	For the gradient runs, the flow rate is set at 2.00 mL/min and the volume fraction of acetonitrile in water increases from 5% to 95%. The two solvent lines containing pure water and pure acetonitrile contain 0.02% (

) trifluoroacetic acid (TFA). The gradient times were set at 1.17 min and 0.10 min, the temperatures at 45 ºC and 70 ºC, the flow rates at 1.0 mL/min and 2.0 mL/min, the detection wavelengths at 210 nm and 257 nm, and the injection volumes at 1 μL and 10 nL when using an Acquity I-Class UHPLC system (Waters) and the reduced volume research prototype system, respectively.

Contribution and Limits of Theories of Chromatography for Column Technology Development

	Plate Theories: The very first theory of mass transport in a chromatographic column was proposed by Martin and Synge. Their well-known plate theory for a two liquid-phase system (8) saw the birth of partition chromatography and they both received the Nobel Prize in Chemistry for this discovery in 1952. The theory accounts for band broadening by considering a series of plates of

	equal height in which equilibrium always applies. By analogy to the distillation process, the smaller the plate height is, the larger the number of chemical fractions to be separated and the larger the column efficiency. However, for either open tubular columns (OTCs) or particulate columns, the notion of plates is remote from the reality because it does not relate directly to the physical parameters controlling analyte dispersion. As a result, the plate theory is merely lumping all the phenomena of band broadening into a single and apparent height, 

, which carries no clear physical relevance for the user, and so, the plate theory is not really helpful for column development.

 For the reasons mentioned previously, the so-called rate theories flourished and were hugely successful in the 1950s, the golden decade of chromatography (9,10). Indeed, rate theories considered relevant dispersion parameters (such as the bulk diffusion coefficient 

 between the stationary and mobile phase, and the equilibrium constant 

) and provided explicit expressions for the peak or breakthrough concentration profiles as a function of these parameters. Both chemists and analysts are now disposing of a solid physical model for performance optimization by selecting the most appropriate combination of speed, axial dispersion (particle size or open-tube column internal diameter [i.d.]), and diffusivity in the stationary phase (temperature, particle size, particle porosity, pore size). Van Deemter (11) was the first to unify the plate and rate theories in 1956. He derived his well-known equation in which H became an explicit function of U and three main coefficients related to longitudinal diffusion (called 

), and mass transfer resistance in the stationary phase (called 

 received a physical meaning and became an unambiguous function of 

	Unfortunately, the experience shows that any attempt to fit the van Deemter equation to experimental 

 data always returned meaningless parameters, for example, diffusion coefficients, which are significantly different from the actual ones (12). This sets the limits of rate theories as being an oversimplification of the actual separation process.

 To further bridge the gap between theory and observation, in 1955 Giddings and Eyring elaborated the very first probabilistic or stochastic theory of chromatography (13), assuming random adsorption and desorption events for each individual analyte molecule (such adsorption and desorption events were assumed to follow a Poisson distribution). This early stochastic theory was not fully comprehensive because it did not account for the known sources of band broadening pertaining to natural diffusion and flow heterogeneity in the chromatographic column. This gave birth to what is considered the most elegant theory of band broadening based on the simple one-dimensional (1D) random walk model, which can quantitatively describe all sources of zone dispersion in a chromatographic column including longitudinal diffusion, trans-channel, short-range inter-channel, long-range inter-channel, and trans-column eddy dispersion, as well as mass transfer resistance as a result of finite diffusivity in the stationary phase and to slow adsorption–desorption (14). In particular, this stochastic model revealed that the dispersion of small molecules in randomly packed beds can be significant at high speed, suggesting the design of more ordered structures to minimize flow heterogeneity in the chromatographic column.

	Even though there cannot be any better theory of mass transport than the stochastic theory of chromatography, it cannot properly account for all the relevant specificities of LC columns and porous materials. For example, neither the true external geometry of the particles, their actual size distribution, the real flow velocity profile across the diameter of a column, nor the actual internal structure of a mesoporous packing material are properly considered by these theories (15). Yet, these column and particle properties control the chromatographic performance and are highly relevant.

Morphology Reconstruction, Multiscale Imaging, and Simulation Techniques: 

The gap between the theories mentioned previously and observation is still important today for the lack of an accurate description of the true morphological features of a column and band broadening mechanism. Even the most elaborate rate or stochastic theories ignore some of the most relevant structural features of a chromatographic column pertaining to the macroporous space (mobile interstitial volume), mesoporous space (stagnant internal volume), and to the distribution or density and diffusivities of the solvent and analyte molecules within a single pore. A complete morphological characterization of the column bed is first needed to make physically relevant predictions for the main column properties (retention and efficiency) (16). This task has been supported by the advances in microscopy (optical and electronic), imaging technologies, and computer resources. Focused ion beam scanning electron microscopy (FIB-SEM) (17) or confocal laser scanning microscopy (CLSM, a few nm resolution) (18) are applied for the reconstruction of the mesoporous space while scanning transmission electron microscopy (STEM, a few Å resolution) (19) enables the accurate reconstruction of the mesoporous space inside the particle. Finally, at the molecular level and within a single pore, either Monte-Carlo (20) or molecular dynamics (21,22) simulations are performed to determine the expected density profiles of both solvent and analyte molecules from the pore wall (of various possible surface chemistries) to the centre of a single mesopore. It is striking to see how complex the solvent and analyte density distributions are when visualizing their many ups and downs in their profiles. They underlie the intrinsic heterogeneous nature of the interface between the pore wall and the bulk pore volume.

	Once the macroporosity, mesoporosity, and molecular distributions in the column are physically reconstructed, retention (mass distribution) and transport properties (molecular diffusivities, flow distribution, hydrodynamic dispersion)  are both evaluated by combining the Lattice Boltzmann method (LBM) for simulation of fluid flow profiles and a random-walk particle-tracking (RWPT) technique for calculating the effective bed diffusivities and hydrodynamic dispersion of the analyte.

	FIB-SEM and STEM microscopies combined altogether with LBM and RWPT approaches provide unprecedented levels of information regarding the chromatographic properties because no system oversimplification is assumed in this approach. They can immediately reveal strengths or weaknesses of any novel structures and surface chemistries selected in research and development. Recently, these techniques have been applied to support method development in size exclusion chromatography (SEC). The macroporous and mesoporous spaces of a 2.1 mm × 150 mm column packed with 1.7-μm fully porous particles were physically reconstructed by FIB-SEM and STEM, respectively. LBM and RWPT approaches were applied to simulate fluid flow and internal/bed diffusivities of polystyrene standards of masses in between 90 and 90 000 Da. These simulations in real bed structures allowed the optimum flow rate (~ 0.04 mL/min), which maximizes the global peak capacity over the entire elution space (see left panel in Figure 1), and the particular flow rate (0.22 mL/min), which ensures a nearly uniform rate of peak capacity (or uniform resolution power) across the entire separation window (see right panel in Figure 1, solid green line), to be determined (23).

Next Generation of Chromatographic Columns and Systems

Ultra-Ultra-High-Pressure Liquid Chromatography (UUHPLC)?:

 Can we reasonably conceive a major breakthrough in column and system technologies by moving from today’s UHPLC (typically based on 10 cm × 2.1 mm columns, particle size 

 = 2.0 μm, 1 kbar maximum pressure, 0.8 cP average eluent viscosity, and 1 mL/min maximum flow rate) to tomorrow’s UUHPLC after further shrinking both the particle size and the column length by a factor of about three (for example, 3-cm-long columns and 

 = 0.7 μm particles)? If yes, which would then be the maximum required pressure and the most appropriate column internal diameter in UUHPLC? These questions are answered in the following paragraphs based on basic knowledge in chromatographic sciences.

 For the analysis of the same class of analyte (same diffusion coefficient), the optimum reduced linear velocity being unchanged in both UHPLC and UUHPLC, the optimum linear velocity is inversely proportional to 

. Additionally, the pressure drop is proportional to column length and inversely proportional to 

. Accordingly, the maximum pressure of the UUHPLC system will then need to be around 7 kbar for 3-cm-long columns packed with 0.7-μm particles.

) of slurry-packed columns with 0.7-μm particles is evaluated as a function of the bed aspect ratio, 

, by assuming that the thickness of the dense wall region is unchanged around 130 μm (17). The calculations are based on a validated stochastic approach for modelling the trans-column eddy dispersion coefficient (24). The results are shown in the left panel in Figure 2. Good performance (h

 ≤ 2) is expected if the column internal diameter is either larger than 2.6 mm (maximum flow rate needed: 3.9 mL/min) or smaller than 350 μm (maximum flow rate needed: 70 μL/min). From a pump technology viewpoint, delivering 3.9 mL/min at 7 kbar is very challenging, therefore only micro-UUHPLC is likely to be a reality in the near future. Second, and independently of the success of the slurry-packing process and LC pump technology up to 7 kbar, the frictional heating power (

) involved at the optimum linear velocity is also evaluated as a function of the bed aspect ratio. The results are given in the right panel in Figure 2. Note a non-monotonous trend because the optimum reduced linear velocity νmin depends on the internal diameter of the column. It varies in between 

 = 1.5 (poor packing quality due to strong wall effects) to 

 = 11 (excellent packing quality, bulk dispersion essentially). In practice, the frictional power 

 at optimum velocity should not exceed a few watts per metre, typically 3 watts/m. Otherwise, at higher 

 values, the column efficiency starts to rapidly drop as a result of excessive radial temperature gradients across the column diameter (25–27). Accordingly, UUHPLC will not suffer from nefarious thermal effects if the column internal diameter either lies in between 400 μm and 700 μm or remains smaller than 85 μm.

	To summarize, micro-ultra-ultra-high-pressure liquid chromatography mass spectrometry (μ-UUHPLC–MS) up to 7 kbar seems to be conceivable only for 0.35 mm × 30 mm columns packed with 0.7-μm particles. The maximum flow rate permitted should be close to 70 μL/min, which will minimize thermal effects and efficiency loss. The analyte dispersion as a result of the injector should also be minimized because the volume variance of the μ-UUHPLC column will not be larger than 10

 = 1. This constitutes a challenging task in terms of system integration in micro-LC systems. Clearly, no other column internal diameter appears to be suitable for UUHPLC without causing severe efficiency loss as a result of either poor column slurry packing or excessive thermal effects. Therefore, the only solution for successful UUHPLC–MS systems will consist in insulating large internal diameter columns in a high vacuum (10

 Torr), as recently demonstrated for narrow-bore columns packed with sub-2-μm particles in UHPLC (27–29).

 3D-printed technologies are extremely attractive in the separation field because they enable the fabrication of perfectly ordered beds (structures with minimum zone dispersion) with a high degree of reproducibility. This topic has already been discussed in previous reports (30–33). To summarize briefly, Figure 3 shows different chromatographic structures 3D-printed by UV curing of acrylonitrile-butadiene-styrene oligomers (30), photo/ion lithography (34), and by two-photon polymerization (right panel) (33). It is remarkable that the build volume (100 cm

) of the 3D-printed structure increases at about the power 3/2 of the feature size (150 → 5 → 1 μm). This means that it is extremely challenging to rapidly build a large volume of macroporous and mesoporous materials with small feature sizes. Structures with sub-5-μm feature size are currently limited to either nano-LC or micro-LC because the throughput of current 3D-printing technologies is too low (33). It will be one of the important separation science challenges of the next decade to reduce 3D-printing resolution down to a few micrometres with high-throughput capacities.

Accelerated Chromatography with 1-cm-Long Columns?

The strong push towards high-throughput LC is constantly coming from the separation and analysis field, and, particularly from the pharmaceutical industry (35). Users desperately need columns and LC systems capable of simultaneously reducing the injector duty cycle time, the dwell time of mixers, the separation time, the column equilibrium time in gradient elution mode, the postcolumn sample dispersion, and the MS data acquisition and processing times while maintaining satisfactory resolution levels. A research prototype system was recently designed and assembled in our laboratory. This system reduces to nearly zero both the dwell volume (< 5 μL) and the extracolumn volume (~ 0.5 μL total system volume, ~ 0.2 μL

 total system volume variance) and is designed to operate columns shorter than 1 cm packed with sub-2-μm particles for high-throughput gradient analyses. For the sake of proprietary information, its precise design cannot be disclosed in this 

 article. Yet, in the next sections, its performance observed under isocratic and gradient conditions will be compared to a UHPLC system with fixed loop injection, 2.8 μL total system volume, 1.6 μL

 The very same isocratic run (65% acetonitrile in water, room temperature) operated at optimum column performance (at a flow rate of 0.45 mL/min) was performed on the UHPLC system (fixed loop needle injection mode) and on the research prototype system. Figure 4 shows the corresponding chromatograms and confirms a high degree of system integration between the eluent preheater, the injection valve, the column, the oven, and the detector when comparing the average overall column efficiency 

 (measured for six small molecules five times and an average retention factor 

 = 1) on a UHPLC system and on the research prototype systems. 

 increases from 750 to more than 1800 after system integration, meaning that the reduced plate height (RPH), 

, decreases by nearly a factor of three from 

 = 3.4 (prototype system). Note that the RPH of a 2.1 mm × 10 cm column packed with the very same material and non-corrected for the extracolumn dispersion of the UHPLC system (2.8 μL total system volume,1.6 μL

 = 2.0 (36). The large difference between the measured RPHs for 1-cm-long (

 = 2.0) columns measured on the same system is essentially caused by the presence of the two 0.21 mm × 1 mm frit (0.7 μL void volume, ~ 0.2 μL

 volume variance each [37]). The band dispersion caused by the same frits has obviously more impact on short than on long columns. The elution volume, 

, of the packed bed alone (intrinsic RPH 

 = 1.2 measured on a 10-cm-long column after removing both frit and system dispersion, total bed porosity 55%, and 

, respectively, for a 1-cm-long column. So, 

 for a 10-cm-long column. Therefore, the overall and observed efficiency, 

, of a 1-cm-long column with and without frit operated on two different instruments can easily be estimated:

 = 8.0 for the 1-cm-long column mounted on the UHPLC sytem with fixed loop, 

 = 7.1 for the UHPLC system in the absence of frits, 

 = 3.5 for the integrated research prototype system, 

 = 2.0 for the integrated research prototype system in the absence of frits.

	The first and third RPH calculations are in very good agreement with the observation (

 = 8.3 and 3.4), and the second and fourth RPH calculations reveal that frits have a severe impact on the efficiency of very short columns and that the apparent efficiency of a 1-cm-long column mounted on the integrated research prototype instrument could be improved by over 70% from 

 = 3090 if there were no frit at both ends of this short column.

	Figure 5 demonstrates that 1-cm- long columns intrinsically suffer from significant sample dispersion caused by the two frits and by border (uneven flow distribution) and wall (uneven flow profile) effects. This causes systematic peak tailing (bottom chromatogram in Figure 5), which almost completely vanishes for longer columns (example: 3 cm, top chromatogram in Figure 5) packed with the same material. For instance, the maximum USP efficiency of a 3-cm-long column measured on the same research prototype “reduced volume” instrument is 11 000 with excellent peak symmetry. However, reducing the column length by a factor of 3 does not reduce the column efficiency by just a factor of 3 as would have been expected for the very same packing quality but by a factor of 5.5 (the maximum efficiency is only around 2000 instead of the expected 3700 plates).

 Figure 6 compares two gradient chromatograms recorded for the very same gradient run (see insert in the top left corner in Figure 6) except for the system configuration. The top chromatogram was recorded on the above-described standard UHPLC system equipped with a 50-μL mixer, while the bottom gradient chromatogram was recorded on the “reduced volume” research prototype system. The temperature was set at 70 °C to allow the operation of a 1-cm-long column at a maximum flow rate of 2 mL/min (total system pressure around 8000 psi). The gradient time 

 was extremely short and set at only 0.1 min. The sample mixture contained small molecules relevant to pharmaceutical analysis. It is striking to observe that some early eluting compounds (for example, compound 2: sulfamethazine) are not even caught by the gradient front before their elution, even for a dwell time as short as 1.5 s. Most compounds that were not well focused at the column head show very poor peak shape because they were first eluted under isocratic conditions and then eluted under gradient conditions. The average peak width was of the order of 1 s, with quite poor resolution between compounds 3 and 4. This peak deformation can be explained by the fact that compounds 3 and 4 are first eluted under isocratic elution then under gradient conditions after they have been caught by the gradient front. In the end, the analysis time was about 15 s.

	In contrast, repeating the very same gradient method but on the “reduced volume” system with nearly zero dwell volume and extracolumn dispersion enabled the run to be completed in just 5 s, with a complete baseline resolution of the six compounds. The average peak width was significantly reduced to only 0.15 s. Overall, Figure 7 illustrates how a chromatographic method initially developed on a standard UHPLC system with a 3-cm-long column can be modified on the “reduced volume” system to shrink the analysis time by more than one order of magnitude while maintaining satisfactory levels of peak resolution for both identification and quantification.

	This article has demonstrated that the past and approximate theories of chromatography based on either plate (Martin and Synge), rate (van Deemter), or stochastic (Giddings) approaches can be successfully refined by combining today’s high spatial resolution visualization techniques (FIB-SEM, STEM) and fluid and molecular modelling tools (LBM, MD, and RWPT). Altogether, these techniques enable the nearly exact physical reconstruction of the actual macropore and mesopore space in a chromatographic column and the accurate prediction of their retention and mass transport properties. Though these tools are currently seen as highly sophisticated for the regular chromatography practitioner and researcher and are expert-driven, they will inevitably play an increasing role in the way research will be conducted in the future for the development of new stationary phase chemistries (optimization of selectivity for particular separation problems) and novel macropore and mesopore structures (performance optimization).

	This article has also revealed that the next and significant improvement in the performance of narrow-bore columns, similar to that which occurred in the mid-2000s from HPLC to UHPLC, is unlikely to be supported in the next decade by either current 3D-printing technologies (for the lack of spatial resolution and high throughput for small feature sizes) or ultra-ultra-high-pressure liquid chromatography (UUHPLC) up to 7 kbar using sub-1-μm fully porous particle FPP or superficially porous particle (SPP) technology in 3-cm-long column format. The fundamental reasons behind such pessimistic views about UUHPLC are that (i) The resolution power of 400-μm- to 1-mm-i.d. slurry-packed columns will not be sufficiently high as a result of inevitable wall effects (38,39) and (ii) Frictional heating and peak distortions will be too excessive for column internal diameters larger than 1 mm. The possible success of narrow-bore column technology above a few kbar will only rely on the full thermal insulation of the column, as was recently demonstrated by applying a high vacuum (10

 Torr) around the column walls (36,37). A major breakthrough in column performance still remains possible without resorting to high-vacuum pumps, but it will be confined to either nano- or at best to micro-LC as demonstrated by Jorgenson with 10-μm- to 150-μm-i.d. capillary columns packed with 1-μm particles (41,42), and by Desmet with 2D-ordered pillar-array columns (32). The remaining barrier for packed column technology will be to adjust the proper agglomeration of sub-1-μm particles and their slurry concentration for optimized packing conditions.

	Finally, it has been demonstrated experimentally that the current gradient separation speed applied in the pharmaceutical industry (based on 3-cm-long columns and conventional UHPLC systems) could easily be accelerated by one order of magnitude by adopting a zero-extracolumn-volume system with short 1 cm × 2.1 mm i.d. columns packed with sub-2-μm SPPs at standard flow rates of 2 mL/min and a high temperature of around 70 °C. The remaining limitations in ultrafast gradient LC are related to the long duty cycle of the injector (move needle or vial, draw sample volume, and wash needle), the poor efficiency of short columns (as a result of border and wall effects and frit dispersion), and the low full scan rates (~10 Hz maximum) and large postcolumn dispersion (~2 μL

 minimum) of easy-to-use quadrupole-based MS detectors for accurate analyte identification and quantification. These current limitations will set the roadmap for future developments in LC instruments in the years to come.

	The author acknowledges Michael Fogwill (Waters Corporation Milford, Massachusetts, USA) for designing and assembling the essential pieces of the research protype “reduced volume” UHPLC instrument, and Sebastien Besner (Waters Corporation, Milford, Massachusetts, USA) for preparing the specific optical detection cell used for the research prototype “reduced volume” UHPLC instrument.

		J.E. MacNair, K.D. Patel, and J.W. Jorgenson, 

		J.J. Kirkland, T.J. Langlois, and J.J. DeStefano, 

		F. Gritti, A. Cavazzini, N. Marchetti, and G. Guiochon, 

		J.R. Mazzeo, U.D. Neue, M. Kele, and R.S. Plumb, 

		T.F. Bruce, T.J. Slonecki, L. Wang, S. Huang, R.R. Powell, and R.K. Marcus, 

		J.J. Van Deemter, F.J. Zuiderberg, and A. Klinkenberg, 

, (Marcel Dekker, New York, New York, USA, 1965).	

		U. Tallarek, D. Hlushkou, J. Rybka, and A. Höltzel, 

		A.E. Reising, S. Schlabach, V. Baranau, D. Stoeckel, and U. Tallarek,

		S.J. Reich, A. Svidrytski, D. Hlushkou, D. Stoeckel, C. Kübel, A. Höltzel, and U. Tallarek, 

		R. Lindsey, J. Rafferty, B. Eggimann, J. Siepmann, and M. Schure, 

		S.M. Melnikov, A. Höltzel, A. Seidel-Morgenstern, and U. Tallarek, 

		J. Rybka, A. Höltzel, A. Steinhoff, and U. Tallarek, 

		F. Gritti, J. Hochstrasser, A. Svidrystski, D. Hlushkou, and U. Tallarek, 

		F. Dolamore, C. Fee, and S. Dimartino, 

		K. Broeckhoven, D. Cabooter, S. Eeltink, W. De Malsche, F. Matheuse, and G. Desmet, 

		W. De Malsche, D. Clicq, V. Verdoold, P. Gzil, G. Desmet, and H. Gardeniers, 

		F. Gritti, S. Shiner, J. Fairchild, and G. Guiochon, 

		K.D. Patel, A.D. Jerkovich, J.C. Link, and J.W. Jorgenson, 

		J. Godinho, A. Reising, U. Tallarek, and J.W. Jorgenson, 

 is currently a Principal Consulting Scientist at Waters Corporation, which he joined in 2015. He received his PhD in chemistry and physics of condensed matter from the University of Bordeaux (France) in 2001 and pursued fundamental research in Prof. Georges Guiochon’s laboratory as a Research Scientist at the University of Tennessee Knoxville (USA) until 2014. Dr. Gritti’s main research interests involve liquid/solid adsorption thermodynamics and mass transfer mechanism in heterogeneous media for characterization and design optimization of new liquid chromatography instrument/columns.

Articles

Features

A series of theoretical, visualization, and simulation tools that are used to improve the structure and chemistry of the next generation of liquid chromatography (LC) columns is briefly reviewed.

Next Generation of Chromatographic Columns and Systems: From Theories to Possible Future Practices

On-line comprehensive two-dimensional liquid chromatography (on-line LC×LC) provides much higher separation power (higher peak capacity) than one-dimensional liquid chromatography (1D-LC). However, it is also often thought that a larger peak capacity should be obtained at the expense of a higher dilution (lower peak intensity). From a theoretical approach, it is demonstrated that both demands can go hand-in-hand in on-line reversed-phase LC×reversed-phase LC (higher peak capacity and higher peak intensity). Examples involving “sub-hour” separations of a tryptic digest show how this approach can be applied in practice.

	As a result of its huge separation power within relatively short analysis times, comprehensive two-dimensional liquid chromatography (LC×LC) has been used in a wide variety of applications over the past ten years. The large number of recent books and reviews dealing with 2D-LC applications is an indication of the attractiveness of this separation technique (1–4). The final goal of LC×LC is to obtain the highest possible separation power (high peak capacity) with the highest possible peak intensity (low dilution), within the shortest possible analysis time. Among these three conflicting goals, the separation power (peak capacity) and the analysis time (the gradient time of the first dimension) are usually considered as the critical quality descriptors, while the dilution is most often disregarded. For the separation of peptides or, more generally, for the separation of ionizable compounds, the use of reversed-phase LC in both dimensions allows impressive peak capacities by simply changing the pH between both dimensions to be achieved (5). For given analysis times, recent studies have compared the peak capacities obtained in 1D-reversed-phase LC against reversed-phase LC×reversed-phase LC. For peptides, it was found that reversed-phase LC×reversed-phase LC outperforms 1D-reversed-phase LC above a total analysis time of 5 min (6). More recently, a peak capacity of 10 000 was achieved in four hours using an active modulation approach (7). Compared to conventional one-dimensional liquid chromatography (1D-LC), LC×LC is usually thought to provide limited performance in terms of sensitivity as a result of solute dilution during the separation process. However, a comprehensive and fair comparison of dilution between these two separation techniques has never been investigated so far. In this article, the concept of dilution in gradient elution is revisited to provide efficient tools to compare in a reliable manner the dilution factors between 1D-reversed-phase LC and reversed-phase LC×reversed-phase LC. The validity of the proposed method is assessed through the separations of peptides in 1D-reversed-phase LC and in reversed-phase LC×reversed-phase LC.

	In gradient elution, the effective peak capacity, 

, the average peak standard deviation in time units, resulting from band broadening into the entire separation system (instrument and column).

	In 1D-LC, the sample dilution depends on both the injection volume and the peak width. Assuming a Gaussian peak shape, the dilution factor, 

, is given by the following relationship:

 is the column peak standard deviation in volume units, and 

, the injected volume. β corrects for additional band broadening due to extra-column dispersion (β = σ

). It may vary from 0 to 1 depending on the importance of extra-column dispersion. Ideally, β should be higher than 0.7 (β² > 0.5) otherwise the loss in plates becomes too significant (>50%) and, furthermore, the assumption of a Gaussian peak shape becomes unacceptable (6). According to equation 1, β represents the fraction of remaining peak capacity. It is given by:

	With x², the ratio of the peak variance, resulting from extra-column dispersion, to the peak variance, resulting from column dispersion (σ

  is the additional extra-column variance due to band broadening in the capillary tubes and in the detector cell, δ

 is a correction factor which depends on the injection process (12 for an ideal injection plug and close to 4 in practice) (8). 

 is the peak injection compression factor: 

  > 1 if the injection plug is narrowed by on-column focusing (injection solvent weaker than the mobile phase), 

  < 1 if the injection plug is broadened when travelling across the column (injection solvent stronger than the mobile phase), and 

 = 1 if the injection solvent corresponds to the mobile phase. In gradient elution, the considered mobile phase is the solvent which accompanies solute elution. Assuming that the injection process is the main cause of extra-column dispersion, σ

 can be considered as close to 0 and then the combination of equations 3 and 4 leads to:

	For a given injection compression factor, equation 5 permits the calculation of the injected volume able to keep a given β value (ideally > 0.7 as mentioned previously). For example, in the situation where the injection solvent corresponds to the elution composition of the first eluted solute, that is, 

 = 1 for this one), the volume that can be injected while keeping 90% or more of the peak capacity for all peaks (exactly 90% for the first eluted one) should be close to the column standard deviation (

	If the injection volume is selected from equation 5, the dilution factor can then be expressed as:

	According to equation 6, the dilution factor should be 2.9 for the least retained compound in the above example (assuming δ = 2). More generally, if the injected volume is selected as a multiple (λ) of the column standard deviation in such a way that β remains very close to 1 (x close to 0), then the dilution factor is close to √2πλ for all peaks in any gradient conditions (equation 2). In most reported studies dealing with the comparison of performance between different separation techniques, the injection volume is either fixed (9) or adapted to the column dead volume (6), but never adapted to the gradient conditions and hence to the column standard deviation. However, this latter approach, based on a proper preliminary selection of λ, is much more attractive because, in this case, dilution and, therefore, peak intensity becomes independent of gradient conditions, that is, flow-rate, column geometry, gradient time, column plate number. Of course, that requires a reliable prediction of the column standard deviation, which is given in gradient elution by:

 is the column dead volume which can be calculated from both the column dimensions and the total column porosity. 

 is the column plate number which can be estimated from the column length, the particle diameter, Knox curve coefficients and predicted diffusion coefficients (10), 

  is the retention factor at the time the solute is eluted, and 

 is the peak compression factor (11), taking into account band compression, especially in the case of very steep gradients. For the rest of this study, 

 will be assumed to be close to 1 in every gradient conditions.   

	For linear solvent strength (LSS) gradients (12) such as linear gradients in reversed-phase liquid chromatography (RPLC) and for large retention factors at initial composition, 

 is the normalized gradient slope (s =âC x 

, with âC the composition range and 

, the column dead time) and S, the slope of the relationship between the logarithm of the retention factor and the eluent composition. It should be noted that the calculation of σ

 makes it also possible to predict the peak capacity in any gradient conditions according to equation 1. Figure 1 shows the calculated kinetic curves for different column lengths. Each curve represents the gradient time as a function of the predicted maximum attainable peak capacity. These curves were calculated for peptides (assuming a β value close to 1) according to the procedure described in reference 6. For a given particle diameter and a given column temperature, each column length, that is, each curve, is related to the highest possible flow-rate considering the maximum allowable pressure (here 1000 bar), thereby leading to the highest column plate number that can be achieved with the resulting column dead time (13–15). It is widely accepted that reducing the particle size and increasing column temperature allows an increase in the column plate number per time unit (16). The curves in Figure 1 were therefore predicted under high-temperature-ultra-high performance liquid chromatography (HT-UHPLC) conditions (sub-2-µm particles, here 1.7 µm were used at a temperature of 80 °C which is the highest column temperature that can be applied to current silica-based columns for a long term period). The limit curve (dotted line in Figure 1) gives the maximum attainable peak capacity for a given gradient time, for example, 530 in 30 min, or the shortest gradient time for a given peak capacity, for example, 3 days for 1500. It is important to insist on the fact that the limit curve to the best existing sets of conditions in 1D-LC, with packed columns, for the separation of peptides. In such HT-UHPLC conditions, the peak capacity seems to tend towards a limit of about 2000. Each circle data point located on the limit curve corresponds to a set of conditions including column length, flow-rate, gradient time, and also the calculated to provide similar peak intensities in all gradient conditions. The arrows in Figure 1 indicate the column lengths and the injection volumes for a λ value of 1. The required injection volume increases with the gradient time for a given column length.

	In addition to be more attractive due to its ability to standardize the dilution in 1D-LC, the proposed approach is, in our opinion, the only way to provide a relevant comparison in term of dilution between 1D-LC and on-line LC×LC. This is of prime importance in the light of the smooth gradients usually applied in the first dimension (

D) which lead to large peaks and hence high dilution when the injected volume is not properly adapted. Injection volumes in 

D calculated in a similar way than as in 1D-LC (same λ value) should lead to similar peak intensities. The dilution factor in LC×LC is usually considered as the product of the dilution factors in each dimension (

 should be close to any dilution factor in 1D-LC (provided that λ is kept identical), the comparison of the dilution between 1D-LC and on-line LC×LC thus becomes only dependent on 

	Equations 1, 2, 3 and 4 can be applied in 

-injection compression factor is high enough to ensure that 

β > 0.7. The injection compression factor is related to two solute retention factors: 

, obtained with the injection solvent as mobile phase, and 

, obtained with the mobile phase at elution. 

 is usually well approximated by the following relationship (10,17,18):

	In the particular case of on-line reversed-phase LC×reversed-phase LC, with 

 values expected to be similar in both dimensions (

	An optimization procedure has been developed for on-line LC×LC separations, based on predictive calculations tools (10). As was shown, very steep gradient in 

 values. The reported procedure allowed to plot the dilution factor as a function of the effective peak capacity for different sets of LC×LC conditions (pareto plots), considering a given gradient time in the first dimension (given analysis time). Figure 2 shows the obtained pareto plots, calculated for four different gradient times (26 min, 40 min, 100 min, and 200 min for a composition range of 35%; all other conditions are given in the figure caption). Each colour is related to a set of conditions (column length, column diameter, and particle size in both dimensions). Each symbol is associated to both a flow-rate value in 

D and a sampling rate value. UHPLC instrumentation usually yields low values for σ

(typically < 10 µL²). However, unlike in 

D and in 1D-LC, its contribution to the total peak variance in 

D may be quite significant (equation 4) and was therefore taken into account in our calculations, that is, 6 µL²). As in Figure 1, a λ value of 1 was used for the calculation of injection volumes in 

  = 2.5 in all studied conditions. Note that the entire fraction, collected at the outlet of 

D (no flow splitting), even when the column diameter was the same in both dimensions, that is, 2.1 mm. The large red data points circles correspond to the best calculated peak capacities in 1D-LC (limit curve in Figure 1). For each gradient time, the pareto-front (blue dotted curve) represents the performance limit in reversed-phase LC×reversed-phase LC with respect to the target objectives, that is, maximizing the peak capacity while minimizing the dilution factor. The large blue circles data points correspond to the three different sets of conditions that were applied to the reversed-phase LC×reversed-phase LC separations of peptides (Table 3) as discussed below. These calculations give rise to some important comments:

		The maximum expected peak capacities are 1250, 1850, 4200, and 7600 in 26 min, 40 min, 100 min, and 200 min, respectively. Interestingly, the dilution factors at the maximum peak capacity are very close to the one predicted in 1D-LC, that is 2.5 (horizontal dotted line). It is important to underline that if post-first-dimension flow splitting was taken into account, pareto optimum curves would be shifted towards even more impressive peak capacities but also higher dilution.

		The dilution factors in LC×LC can be decreased by a factor 5 without sacrificing too much of the peak capacity as highlighted by the red dotted lines.  

		The dilution factors can be further decreased while keeping fairly good peak capacities. For example, a decrease in the dilution factor by a factor of 25 only leads to a decrease of 40% (26 min), 39% (40 min), 31% (100 min), and 28% (200 min) of the peak capacity.

		Among the various conditions evaluated, those involving a short column in 1D (typically 5 cm) and a very short column packed with sub-2-µm particles in 

D (typically 3 cm and 1.7 µm) were always located onto or close to the pareto curve. This is surprising for long analysis times, for example, 200 min, for which longer columns in 

		It is noteworthy that dilution factors are not higher when using a 1-mm column diameter, rather than a 2.1-mm column diameter in 

D. This is confirmed experimentally in the next section.

Acetonitrile (LC–MS grade) was purchased from Sigma-Aldrich (Steinheim, Germany). An Elga Purelab Classic UV purification system from Veolia water STI (Décines-Charpieu, France) was used for water purification and deionization. Formic acid (LC/MS grade), ammonium acetate, and ammonium bicarbonate (both analytical reagent grade) were obtained from Fisher Scientific (Illkirch, France). DL-1,4-dithiothreitol (DTT, 99%) and iodoacetamide (98%) were obtained from Acros Organics (Geel, Belgium). Trypsin, human serum albumin (HSA), bovine serum albumine (BSA), β-casein, myoglobin, lysozyme, and cytochrom C were all obtained from Sigma-Aldrich (Steinheim, Germany). Influenza hemaglutinin (HA), leucine encephalin, bombesin, [arg8]-Vasopressin, [ile]-Angiotensin, bradykinin fragment 1-5, substance P, and bradykinin were obtained from Merck (Molsheim, France). The protein digest sample was obtained by tryptic digestion of six proteins (HSA, BSA, β-casein, myoglobin, lysozyme and cytochrome C) following a protocol described elsewhere (10,19). The ratio enzyme to substrate was 1/70, resulting in a complex sample containing 196 peptides with an average concentration of 120 mg/L.

	The standard sample was obtained by diluting eight peptides in pure water at a concentration of 32 mg/L. Their characteristics are listed in Table 1.

	Both samples were filtered on a 0.22-µm PVDF (polyvinylidene) membrane before injection.

 2D-LC experiments were performed on a 1290 Infinity series 2D-LC system from Agilent Technologies (Waldbronn, Germany). The system includes: two high-pressure binary solvent delivery pumps, an autosampler with a flow-through needle injector, two thermostated column compartments with low-dispersion preheaters, and two diode-array detectors (DAD) with 0.6 µL flow-cells. The interface connecting the two dimensions was a 2-position/4-port duo valve, equipped with two identical 80-µL loops. The loops were emptied in back-flush mode to minimize band dispersion. A pressure release kit was placed between 

D outlet and the interface inlet to minimize the pressure downstream coming from the valve switching. The measured dwell volumes and extra-column volumes were respectively 140 µL and 22 µL in 

D (loop volumes excluded). For 1D-LC experiments, only the first dimension of this system was used. In both cases, the system was hyphenated to a quadrupole-time-of-flight (QTOF) high resolution mass spectrometer (model G6560B) from Agilent Technologies, equipped with a JetStream electrospray ionization (ESI) source. As a result of the very high flow rates operated in 

D was split (1:2) before entering the source. Instrument controlling was performed with OpenLab CDS Chemstration software (Agilent) for the 2D-LC-UV system, and MassHunter software (Agilent) for the mass spectrometer. UV data were acquired at 210 nm with an acquisition rate of 40 Hz in 1D-LC, 20 Hz in 

D. High resolution mass spectrometry (HRMS) data were acquired in positive ion mode from 100 to 3200 Da, with an acquisition rate of 20 spectra/s. The operational parameters for the QTOF instrument were the following: drying gas temperature (300 °C), drying gas flow (11 L/min), nebulizer gas pressure (40 psi), sheath gas temperature (350 °C), sheath gas flow rate (11 L/min), capillary voltage (3500 V), nozzle voltage (300 V), fragmentor voltage (150 V), Oct 1 RV voltage (750 V), respectively. HRMS data were processed with MassHunter qualitative analysis software (Agilent).

	In 1D-reversed-phase LC, the separations were performed on two 1.7 µm Acquity CSH C18 columns (Waters). The column dimensions were either 50 mm × 2.1 mm, or 150 mm × 2.1 mm. The mobile phase was composed of water with 0.1% formic acid as solvent A, and acetonitrile with 0.1% formic acid as solvent B (pH = 2.7). The gradient elution was performed from 1% B to 35% B with different gradient times. The gradient time was followed by a quick return to initial eluent composition (1t0) and a short column equilibration time (5t0).The several sets of conditions that were used are given in Table 2.

	Experimental conditions for the three LC×LC set-ups are given in Table 3.

	1D-LC–UV and 2D-LC–UV data were processed using Excel and an in-house script written under MATLAB 7.7. The 

 values for the eight peptides (listed in Table 2) were determined from a HPLC modelling software (OSIRIS 4.2, Euradif, Grenoble, France) from two gradient data (10 min and 30 min).

	The objective of this practical approach was to assess to what extent experimental results are consistent with the calculations above.

	A series of ten gradient runs was carried out in 1D-LC with conditions close to those given in Figure 1 (located onto the limit curve). Ten different sets of gradient conditions (listed in Table 2) were applied to a sample of eight peptide standards and to a tryptic digest of 6 proteins. UV detection was used to compare the peak heights while HRMS detection was used to identify the different peaks. Figure 3 shows the UV-chromatograms obtained with the two samples with four different sets of conditions. As expected, the peak intensities seem to be very similar for a given sample regardless of the gradient conditions. Table 2 lists, for four peptide standards, the measured peak heights obtained with the ten sets of conditions. As highlighted by RSD values, that is, around 10%, the change in gradient conditions has very little impact on peak heights, thereby demonstrating that injection volumes were properly determined from the predicted peak width. Furthermore, good peak shapes could be obtained for both samples, even for the least retained peptides, suggesting that our preliminary assumption (β values close to 1 for all peaks with λ = 

 = 1) was correct. Particular attention should be paid to the gradient conditions on the right side of Table 2 (see also Figures 3[g] and 3[h]). These conditions are very close to those usually applied to a first dimension in on-line reversed-phase LC×reversed-phase LC. They include a short column, that is 5 cm, a long gradient time, that is 40 min, a column temperature of 30 °C, and a flow-rate well below the highest possible one, that is, 0.5 mL/min versus 1.2 mL/min. Such conditions imply very large peak widths (high 

 values) and hence large injection volumes (17 µL, that is about 16% 

). Despite this, peak intensities remain very similar to those obtained in optimum 1D-LC conditions. Our first assumption was that, for any gradient conditions in 1D-reversed-phase LC, it should be possible to keep similar peak heights by injecting the appropriate calculated volume. This assumption seems to be validated by this experimental study.

	Our second assumption was that the dilution in on-line reversed-phase LC ×reversed-phase LC should be greatly reduced compared to 1D-reversed-phase LC. This is demonstrated in Figure 4, showing on-line reversed-phase LC×reversed-phase LC chromatograms for the tryptic digest of six proteins. Three different setups were selected with conditions listed in Table 3. The predicted performance of these setups can be seen in Figures 2(a) (setup #3), and 2(b) (setups #1 and #2). In Figure 4, the contour plots (top figure) show the retention space while the 3D-chromatograms (bottom figure) reveal the peak intensities. The separation quality attributes are given in Figure 4 at top left of the contour plots. They include analysis time, obtained effective peak capacity, percentage of surface coverage, peak widths in each dimension, and number of runs in the second dimension. As can be seen, the obtained effective peak capacities are very close to (slightly lower than) those predicted in Figure 2. They were found to be 2.5 times (in 26 min) and 3.5 times (in 40 min) higher than in 1D-reversed-phase LC (Figures 3(d) and 3(f) respectively). More important is the increase in peak intensity (reported in Table 4) for both studied samples. Peak intensities were multiplied by an average factor of 3.3 to 4.5 depending on the setup. Such increase is in very good agreement with the predictions shown in Figure 2 (a predicted factor of 5). It can also be noted that setups #1 and #2 which differ from the column diameter in 

D (that is, 2.1 mm and 1 mm respectively), lead to quite similar peak capacities (no additional band broadening with 2.1 mm) and quite similar peak intensities (no additional dilution with 1 mm). The number of runs in 

D could be the final quality attribute (120 with 1 mm versus 166 with 2.1 mm) to define the best set of conditions.  

	From both a theoretical approach and an experimental study on peptides, it has been shown that similar dilution factors and hence similar peak heights can be obtained in any 1D-reversed-phase LC gradient conditions, by injecting the appropriate volume, based on a multiple of the column standard deviation.

	This approach, applied to the first dimension of on-line reversed-phase LC×reversed-phase LC, allows an unbiased comparison between the dilution factors obtained in 1D-reversed-phase LC and those in on-line reversed-phase LC×reversed-phase LC. As a result of an important focusing effect in the second dimension (high CF values), very large volumes can be injected, thereby significantly reducing the dilution factor in the second dimension and hence the dilution factor in reversed-phase LC×reversed-phase LC, even when the column diameter is smaller in the first dimension, for example, 1 mm versus 2.1 mm. According to our calculations, the dilution factor might be reduced by a factor of up to 25 in on-line reversed-phase LC×reversed-phase LC while keeping an impressive peak capacity compared to 1D-reversed-phase LC. This seems to be especially attractive for long analysis times (> one hour).  

	For sub-hour separations of a protein digest, our experimental results were in perfect agreement with the predicted ones. For an analysis time of 40 min, the peak intensities could be increased by a factor close to 5 in reversed-phase LC×reversed-phase LC while the peak capacity was increased by a factor of 3 compared to the optimum 1D-reversed-phase LC separation.

		S. Chapel, F. Rouvière, M. Sarrut, and S. Heinisch, 

Two-Dimensional Liquid Chromatography Coupled to High-Resolution Mass Spectrometry for the Analysis of ADCsin Antibody-Drug Conjugates: Methods and Protocols

 (ed. Tumey, L. N.), 163–185 (Springer US, New York, NY, 2020). doi:10.1007/978-1-4939-9929-3_11

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		A. D’Attoma, C. Grivel, and S. Heinisch, 

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		J.C. Sternberg, in: J.C. Giddings, R.A. Keller (Eds.), 

, vol. 2, Dekker, New York, 1966, p. 205	

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A Theoretical and Practical Approach to Manage High Peak Capacity and Low Dilution in On-line Comprehensive Reversed-Phase LC × Reversed-Phase LC: A Comparison with 1D-Reversed-Phase LC

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