The best structures of LC columns-A theoretical perspective.

The largest peak capacity (n) that LC analysis can generate in isocratic or gradient elution analysis of a given sample in a given time at a given pressure is proportional to the quality factor (qmax) of its column structure. In this study, the multi-channel structures with open pseudo-planar channels (OPPC) and open circular channels (OCC) where compared with PC2 - a typical core-shell column packed with 2 µm particles. These columns have qmax of 1.27, 1.17 and 0.41, respectively. The former two qmax are the highest among all known column structures - about 3 times higher than qmax of PC2. This means that the OPPC and OCC can generate about 3 times higher n compared to what a PC2 can in the same analysis time (tanal) at the same pressure, or they require about 81 times shorter tanal (81 is the 4th power of 3) to generate the same n as a PC2 can at the same pressure. However, while PC2 is a commercially available column, there are substantial challenges in manufacturing the OPPC and OCC that can compete with PC2 in practical applications. In order to be competitive with PC2, the OPPC and OCC should have sub-1µm characteristic dimensions (e.g., the inter-pillar distance, g, in OPPC-based pillar array columns, internal diameters of OCC). Thus, in order to compete with PC2 in one scenario, an OPPC requires g ≤ 0.14 µm. Additionally, to be competitive with PC2, OPPC and OCC should be able to sustain the same high pressure. Highlighting the challenges of their design and manufacturing might help to develop the manufacturable columns substantially superior to the packed ones.

Theory of linear focusing in chromatographic columns with exponential retention. Part 2: Analysis of special cases.

The unified approach to studies of different separation techniques (GC, LC, etc.) adopted in Part 1 continued herein. As before, the column temperature in GC, the solvent composition in LC, etc., are represented by the concept of solute mobilization (y). General equations for dynamic (moving) linear y-gradients (gradients in y) in non-uniform columns developed in Part 1 are reduced herein to special cases of uniform columns. Only the uniform columns, the ideal sample introduction, the positive y-rates, the negative y-gradients, and the eluting solutes are considered. Equations for solute band compression, peak width, peak focusing, peak separation and others derived. All equations are expressed via dimensionless parameters eliminating unessential factors like the mobile phase type, column type and dimensions, etc. The conventional gradient elution LC is treated as a special case of the separations with y-gradients. Effects of the y-gradients on solute retention time, on band compression, on peak focusing and on peak separation are analyzed. The equations developed herein, while describing simple and familiar concepts (retention time, peak width, etc.), are, unfortunately, cumbersome in many cases. The good news is that the equations are exact and the conditions of existence of the equations are explicitly formulated so that there is no question where the equations can and where thy cannot be used.

Dimensionless plate height is unsuitable for comparing chromatographic columns.

Dimensionless plate height (h) is differently defined for the columns with different type of internal filling (internal support structure). It is h = H/d for open columns (d is the internal diameter), h = H/dp for packed columns (dp is the particle size), h = H/ddom (ddom is the domain size) for columns with other filling (e.g., monolithic, pillar array). The plate height (H) and the column kinetic performance in general depend on the flow channels through the column filling - on their cross-sectional dimensions (dchan), shape, and wall porosity. Other than through their effect on the flow channels, the dimensions of skeleton of the filling and the ddom do not affect H. Parameters d, dp and ddom cannot be found from external measurements of a column as a "black box" separation device and, therefore, are unmeasurable subjective metrics, and, therefore, so is h. Moreover, in some fillings (e.g., monolithic, pillar array) ddom can be chosen independently of dchan. By increasing ddom without changing dchan, one can make h as small as desired with no effect on H and on the column kinetic performance in general. This implies that h does not represent column performance and cannot be used for comparing differently structured column. In this report, previously introduced measurable (objective) performance metrics - the kinetic performance factor (q) and the quality factor (qmax) are described and evaluated. The latter is suitable for comparing differently structured columns. A column practically achievable performance depends not only on qmax, but on the largest pressure that a column can tolerate, the narrowest flow channels that can be manufactured and other practical factors. Practically achievable performance of several column types is compared in this report.

Theory of linear focusing in chromatographic columns with exponential retention. Part 1: Basic solutions.

This report is the first of 2-part study of the effect of gradients in column parameters on the column performance. If t, x and p are, respectively, time since sample introduction, distance from column inlet and some parameter of solute migration along the column then ∂p/∂t and ∂p/∂x are, respectively, the rate of changing p and the gradient of p. Unified approach to study of gradients and rates in different chromatographic techniques (LC, GC, etc.) has been developed. To facilitate a unified approach, the umbrella term mobilization (y) representing column temperature (T) in GC, solvent composition (ϕ) in LC, etc. is introduced. Differential equations for migration of a solute band (collection of solute molecules) under the following conditions are formulated and solved:The key solutions describe the time of migration of a solute band and the band width - both as functions of the distance traveled by the band. The solutions are used in Part 2 for the study of the effects of the negative gradients in y on column performance in several practically important cases. A reduction of the key general solutions to much simpler equations for gradient LC has been demonstrated herein as an example.

Peak width in liquid chromatography with exponential retention and linear program preceded by isocratic hold.

Expression for peak width in gradient elution liquid chromatography with exponential dependence of solute retention on linearly programmed solvent composition preceded by initial isocratic hold has been derived. A special case of previously defined balanced hold has been considered and compared with published results.

Transport diameters of liquid chromatography columns.

The transport diameter (dM) of a chromatographic column is an objective (measurable) representation of cross-sectional dimensions of the column flow-through channels, and of the effect of porosity of the column internal support structure on velocity of transporting solvent molecules along the column. A column kinetic performance is trade-off between the column separation performance, analysis time and pressure. The dM is a key factor in definition of the kinetic performance factor - an objective metric of a column kinetic performance - and of other objective performance metrics. General equations for evaluation of dM are developed. The dM of PLOT, particulate and pillar-array columns are evaluated and discussed.

Chromatographic parameters: Characteristic parameters of solute retention - an insightful description of column properties.

Previously introduced characteristic parameters of solute retention - the solute characteristic temperature (Tchar) and thermal constant (θ) in GC as well as the characteristic solvent composition (ϕchar) and the composition-constant (Φ) in LC - are described herein from a single perspective. Unlike conventional thermodynamic parameters, e.g., entropy and enthalpy of solute evaporation from stationary phase in GC that do not have a direct chromatographic meaning, the characteristic parameters do. Thus, Tchar and ϕchar are the column temperature and the solvent composition in GC and LC, respectively, at which the solute retention factor (k) is equal to 1. Quantities θ and Φ in GC and LC, respectively, are analogous to the time-constant in scientific and technical fields. Thus, e.g., Φ is the increase in the solvent composition leading to e-fold reduction (e ≈ 2.72) in k. Properties of characteristic parameters illustrating intuitive nature and simplicity of based on them concepts and equations are described.

Methodology of quantitative comparison of practically achievable kinetic performance of differently structured liquid chromatography columns.

Previously introduced structural quality factor (qmax) of LC column is a measure of its potential kinetic performance - the larger is qmax the proportionally larger is peak capacity (nc) of isothermal and gradient analysis with the same analysis time (tanal) and pressure (Δp). However, the best practically achievable column performance depends not only on qmax, but is limited by the narrowest practically achievable widths of the flow-through channels in the column internal support structure. These widths can be represented by the smallest practically achievable column permeability (Kmin). Large Kmin might result in unnecessarily high nc in prohibitively long tanal. Structure-independent methodology of evaluation of these effects on a column performance limits and relevant equations are developed. The best performance (the shortest tanal at a given nc) is achieved when a column operates at its highest acceptable Δp (Δpmax) even if (for reducing tanal) the column has to operate under non-optimal conditions. To demonstrate the methodology, three column types with substantially different qmax - the solid-core particle columns (SCPCs), qmax=0.4; the pillar-array columns (PACs), qmax=0.65, inter-pillar distance 1.25 µm; and the porous-layer open-tubular columns (PLOTCs), qmax=0.97, 4.6 µm ID - were compared. It is shown that, because practically available Kmin of SCPCs is much smaller than that in evaluated PLOTCs or PACs, the latter two cannot outperform contemporary SCPCs with dp≤4µm in applications with practically acceptable tanal, although SCPCs have the lowest qmax. Factors affecting Kmin, and the effect of Δpmax on a column performance limit were also evaluated and discussed.

Column length is a structure-independent measure of solvent consumption in liquid chromatography.

Column length (L) is a measure of solvent consumption in LC analysis. In its latter role, L is the specific solvent consumption - the void time (tM) solvent consumption per unit of the column flow-area (AF) which is the cross-sectional area of the column open space (external and internal pores). In tanal-long LC analysis (isocratic or gradient), the solvent consumption (VS) is VS = AFLtanal /tM regardless of flow rate (F) as long as all changes in column dimensions and operational parameters are translatable (in gradient analysis, the ratio of the gradient time (tG) to tM remains fixed).

Measurement of optimal flow rate in gradient elution liquid chromatography.

Method development in gradient LC relies upon the selection of a solvent time program and a mobile phase flow rate. The flow rate, optimal for gradient separation cannot be inherently predicted by the isocratic value optimal for a given analyte, and rather should be identified independently to ensure the highest separation performance of gradient analysis. The optimal flow rate (Fopt) is defined herein as the solvent volumetric flow rate (F) maximizing the separation (Δs) of a predetermined peak-pair or the separation capacity (sc) of the entire LC analysis. The theoretical background and the experimental technique of measurement of Fopt in gradient elution analysis were considered and experimentally demonstrated. The technique of measuring Fopt is based on translatable changes of F where the product FtG (tG is the gradient time) was the same for all values of F. The Fopt was found as F corresponding to the maximum in Δs or in sc.

Optimal heating rate in constant pressure and constant flow gas chromatography.

Optimal heating rate is the one resulting in the shortest analysis time for achieving a required separation performance of a column. The previously recommended default heating rate (RT,Def ) was optimal for temperature-programmed gas chromatography analyses in constant pressure mode. It has been shown herein that the same recommendation can be extended to constant flow mode with fixed heating rate (RT ). The numerical value of RT,Def has been herein rescaled from previous 10 ∘ C / t M (10°C per void time) where tM was measured at 50°C, to 12 ∘ C / t M with tM measured at 150°C-a round number in the middle of the gas chromatography temperature range, chosen as a reference temperature for numerical values of all temperature-dependent gas chromatography parameters. It has been experimentally found based on theory developed herein that R T , Def = 12 ∘ C / t M is optimal for columns with φ = 0.001 ( φ = d f / d is dimensionless film thickness, d and df  are the column internal diameter and film thickness, respectively) in constant pressure mode and constant flow mode with fixed RT . Theory shows that, for arbitrary φ, R T , Def = 12 ( 1000 φ ) 0.09 ∘ C / t M . The theory also shows that the fixed RT is optimal for constant pressure mode. In constant flow mode, however, the optimal RT should gradually increase with increasing temperature (T). The optimal theoretical curves RT (T), different for different flow rates, were found. However, only the optimization of the fixed RT was experimentally evaluated due to limited capability of existing gas chromatography instrumentation and resources. It has been shown that the separation-time tradeoff in constant pressure mode is slightly better than that in constant flow mode. The experimental data are compiled in the Supporting information.

Basic Structure-Independent Equations of Kinetic Performance of Columns in Liquid Chromatography.

The lowest dimensionless plate height (hmin) of the liquid chromatography (LC) column is a subjective metric that cannot be found from measurements of parameters of a column as a separation device and is not suitable for comparison of kinetic performance of differently structured columns. In some cases (monolithic, pillar-array columns), there is no correlation between hmin (as it is currently understood) and the column performance. The same is true for the flow resistance parameter (ϕ). Recently introduced measurable effective diameter and structural quality factor (qmax) of a column are objective replacements for ϕ and hmin. Metric qmax, the maximum of the flow-dependent kinetic performance factor (q), is suitable for comparison of differently structured columns. Structure-independent basic equations binding kinetic performance of LC column with its q and other parameters and operational conditions were developed. It has been shown that previously known and new equations of a column kinetic performance can be derived from the basic ones. An example of using the equations for solving a known practical problem of column selection is provided.

Simulation of the effects of negative thermal gradients on separation performance of gas chromatography.

The effect of a gradient of solute velocity on the chromatographic separation of closely spaced solutes is investigated by usage of a simulation. The concept of the ideal basic separation (IBS), introduced by Blumberg, is used to determine the theoretical limit of a separation without any natural or artificial gradients of features of the chromatographic medium. The IBS is the best achievable separation and can therefore be used as a baseline to which other separations can be compared to. An addition of a negative velocity gradient cannot improve the separation of closely spaced solutes. The velocity gradient is realized by adding a temperature gradient to a GC separation. The simulation confirms this theoretical limit for a range of differently strong retained solutes. In a second part controlled deviations from IBS are used to show, that a velocity gradient can be beneficial in realistic, non-IBS. The addition of a negative velocity gradient can improve e.g. the separation of broad injected solute zones or counteract a positive gradient of the mobile phase velocity caused by gas decompression along the GC column. However, the improved separation cannot exceed that of a corresponding ideal basic separation. The resolution of broadly injected solutes can be increased by up to 45% of the corresponding IBS resolution by adding a negative velocity gradient. A positive velocity gradient due to gas decompression reduces the separation by up to 6%. The added negative velocity gradient, realized by a linear temperature gradient, can compensate this resolution loss by up to 2%.

Practical limits to column performance in liquid chromatography - Optimal operations.

Columns of different structures have different potential kinetic performance - the trade-off between separation, time, and pressure. However, the full potential of a structure cannot always be realized in practically existing columns. Each combination of column efficiency, time, and pressure requires certain cross-sectional dimensions of the column flow-through channels. However, there are limits to the narrowest flow-through channels that can be manufactured with current technology. As a result, columns of some structures cannot be optimized for providing the required efficiency in the shortest time. Additionally, the full potential of its structure can be realized only if a column can operate at the highest pressure available from liquid chromatography (LC) equipment, has sufficient loadability, and satisfies other practical requirements. Equations tailored for a systematic approach to evaluation of factors affecting performance of optimized LC columns (effects of column structure, column dimensions, operational conditions, etc.) were developed. Parameters quantifying the performance of a specific column at and below its largest acceptable pressure were identified. New objective performance parameters of columns and their structures were introduced. Among them are the apparent structural quality factor accounting for the effect of insufficiently high pressure acceptable for the column, the dimensionless plate duration - the parameter of a column structure affecting its performance when the pressure is not limited, - and others. Applying the theory developed herein to published data, the performance of several differently structured columns is evaluated, and the factors affecting their comparative performance are discussed. In the final count, not the quality of a column structure, but practical factors such as the narrowest manufacturable flow-through channels can dominate the choice of the kinetically most suitable column for a practical LC analysis.

Kinetic performance factor - a proportional metric for comparing performance of differently structured liguid chromatography columns.

Particle size (dp) of a column packing material is a subjective quantity, especially in open-tubular, monolithic, pillar-array and others non-particulate columns. It is possible to design a column internal structure so that the dimensionless plate height, h=H/dp, could be arbitrarily small without improving the column performance. Subsequently, h is unsuitable for comparing performance of differently structured columns. The problem is avoided if previously introduced effective diameter (deff) of a column - a measurable (objective) parameter - is used instead of dp. Previously introduced general definition of deff was suitable for LC with incompressible solvents and GC with compressible carrier gases. The latter feature complicates the concept of deff and its relation to other column parameters. Only LC with incompressible solvents is considered in this report. This substantially simplifies the concept of deff and objective performance metrics related to deff, including hc=H/deff and kinetic performance factor (q) proportional to (1/hc)1/2. Moreover, the Bristow-Knox separation impedance (E) - also an objective metric - can be expressed as [Formula: see text] . Metrics q, hc and E carry the same information, the ratio H/deff, only differently scaled and differently powered. However, q is the only metric proportional to external metrics like the peak capacity. Open-tubular columns (OTC) have the highest qmax among all other known structures. Having qmax close to one, OTC are convenient performance benchmark for other structures. Deficiencies of parameter h are demonstrated, properties of parameters deff and qmax are discussed and their numerical values found from published data for several column structures are compared.

Simulation of spatial thermal gradient gas chromatography.

A model to simulate the gas chromatographic separation in the presence of a spatial thermal gradient is presented. This model is developed from existing models for the prediction of retention times in temperature programmed GC. It is based on basic fluid mechanics of gasses in capillaries to describe the properties of the mobile phase and a thermodynamic model to describe retention of solutes in a stationary phase. This model is expanded to incorporate a spatial thermal gradient. The development of the peak width during the chromatographic separation is modeled by a differential equation, which uses the solute residency, the inverse of the solute velocity, instead of the solute velocity. The presented model is compared to measurements of n-alkanes with conventional temperature programmed GC-FID and to measurements with a hyper-fast flow-field thermal gradient GC (FF-TG-GC) coupled with a MS. The FF-TG-GC enables the use of a spatial thermal gradients. For temperature programmed GC-FID, without spatial thermal gradients, calculated retention times are mostly within 1% of measured values. For the FF-TG-GC-MS with a thermal gradient the simulation showed a deviation of the spatial thermal gradient from a linear to a nonlinear gradient, which could be confirmed by measuring the shape of the spatial gradient. The calculated retention times for the nonlinear gradient are also mostly within 1% of measured values. Calculated peak widths are smaller than measured peak widths by 10 to 15% in the case of the conventional GC-FID and by 30 to 50% for the FF-TG-GC-MS. The relation between the measured and calculated variances shows a linear correlation which can be used to correct the calculated variance and peak width. With this correction the difference for the peak widths is reduced to 4-10% for the conventional GC and below 10% for the majority of solutes with exceptions for early and late eluted n-alkanes (up to 25% difference).

Equation for evolution of temporal width of a solute band migrating in chromatographic column.

An alternative differential equation to model the development of the temporal width of a solute band during its migration in a chromatographic column is developed. This model uses the solute residency, the inverse of the solute velocity, as parameter, which has the advantage to be an always finite quantity even at the column outlet in GC-MS where carrier gas velocity approaches infinity. The new differential equation is derived from a known equation describing the evolution of the spatial variance of migrating band. Supplemental material illustrating solutions of the newly developed equation in several special cases is provided.

Thermodynamics-based retention maps to guide column choices for comprehensive multi-dimensional gas chromatography.

Comprehensive two-dimensional gas chromatography (GC × GC) provides enhanced separation power over its one-dimensional counterpart - gas chromatography (GC). This enhancement is achieved by the inclusion of a secondary column, the choice of which is a major determinant on the quality of the ultimate separation. When developing and optimizing a new GC × GC method, the choices of stationary phase chemistries, geometries and configurations (which phase serves in which dimension) are of fundamental importance, and must often be addressed even before the manipulation of instrumental conditions. These choices are often made using educated guesses, literature searches, or trial and error. Thermodynamic models of GC separations; however, provide a fast and easy means of acquiring information for guiding these choices. By using characteristic thermodynamic parameters (characteristic temperatures, Tchar, and characteristic thermal constants, θchar), we demonstrate the generation of maps that can inform the choices of column chemistries, phase ratios and configurations for GC × GC separations.

Migration and elution equations in gradient liquid chromatography.

What is known in the literature as the fundamental equation for gradient elution (FEGE) was previously proven only for conventional gradient LC - uniform (the same at any distance from the inlet) static (fixed in time) solvent velocity (um) in a column of uniform and static internal structure, cross-section and thermodynamic properties. A published alternative to the FEGE - the general migration equation - is valid for any column-based linear chromatography (GC, LC, SFC etc.). It allows one to theoretically or numerically predict a solute migration time to any location along the column. Starting from that general equation, several migration equations in gradient LC under different operational conditions including non-uniform non-static um, Neue-Kuss retention model and others have been developed in this report. It has been shown that the conditions of validity of the FEGE can be expanded to include non-uniform um. On the other hand, the FEGE is not valid for other unconventional operations of LC including gradient LC with dynamic (changing in time) um. This implies that FEGE cannot be applied to, e.g., gradient LC operating at constant pressure where, due to the change in solvent composition, the solvent viscosity changes causing the change in um with time. Applications of newly developed equations to other unconventional operations of gradient LC were also considered. Several new time parameters of the mobile phase flow were identified, interpreted, and evaluated.

Kinetic performance factor - A measurable metric of separation-time-pressure tradeoff in liquid and gas chromatography.

A measurable dimensionless metric of tradeoff between separation, time and pressure in chromatography - the kinetic performance factor (ec) is introduced. Measurable the same way for all LC and GC columns (open-tubular, packed, monolithic, pillar-array, etc.) regardless of retention mechanism, ec supersedes metrics of a column structural quality such as dimensionless plate height (h), Bristow-Knox separation impedance (SI), Golay specific performance index (SPI) and others. Peak capacity, and other separation performance measures of a chromatographic system are proportional to ec. To find ec of a column, there is no need to know the column unmeasurable parameters such as h, stationary phase porosity, resistance factor, and others. To theoretically predict the separation-time performance of a particular column with known ec, two measurable parameters - a column effective diameter (deff) and nominal diffusivity (Dn) were introduced in addition to ec. Quantities ec and Dn represent the properties of column classes while deff represents a particular column. The use of parameters deff, ec and Dn for evaluation of relations between efficiency and void time (tM) in columns of several types has been demonstrated. It is always more time-efficient to use maximum available instrumental pressure (Δpmax) for obtaining a predetermined column efficiency even if that efficiency is relatively low so that operation at Δpmax requires non-optimal flow. It has been also demonstrated that reducing the characteristic cross-sectional dimension (deff, particle size of a packed column, internal diameter of and open-tubular column, etc.) of a column operating at the same Δpmax reduces the column efficiency with the benefit of much larger reduction in the analysis time.