A comparison of 2 micron inner diameter open tubular column liquid chromatography with pressure-driven isocratic, slip-flow, and electrochromatographic modes of operation: a theoretical study.

Modifications to the flow profile used in open tube capillary liquid chromatography (OT-CLC) include using slip-flow walls and using electroosmosis as a fluid pump as practiced in electrochromatography. These modifications are implemented experimentally by changing the capillary surface and solvent conditions which results in the change of boundary conditions at the capillary wall. In this paper we employ a theory-based study and compare the zone broadening of simple solutes using parabolic flow from a liquid pump, slip-flow from a highly hydrophobic inner surface with water eluent, and electroosmosis for the conditions of pure water and dilute salt utilizing 2 µm inner diameter OT capillaries. In general, the two types of flow other than parabolic exhibit thin zones in the early part of the chromatogram, consistent with previous studies of slip-flow and electroosmotic flow used in electrochromatography. Electrochromatography is shown to yield higher efficiency and less zone broadening than parabolic and slip-flow conditions used in this study. Nonetheless, it is found that the zone standard deviations are shown to be similar for these flow profiles as is the number of plates for these different flow profiles under the conditions utilized here. It is revealed that these modifications do not warrant the effort to maintain the special solvent conditions when compared to gradient elution OT-CLC, which gives a nearly constant peak width throughout the chromatogram, is easiest to implement, and is the method of choice for complex analysis.

Estimation of low-level components lost through chromatographic separations with finite detection limits.

The search for biomarkers allowing the assessment of disease by early diagnosis is facilitated by liquid chromatography. However, it is not clear how many components are lost due to being present in concentrations below the detection limit and/or being obscured by chromatographic peak overlap. First, we extend the study of missing components undertaken by Enke and Nagels, who employed the log-normal probability density function (pdf) for the distribution of signal intensities (and concentrations) of three mixtures. The Weibull and exponential pdfs, which have a higher probability of small-concentration components than the log-normal pdf, are also investigated. Results show that assessments of the loss of low-intensity signals by curve fitting are ambiguous. Next, we simulate synthetic chromatograms to compare the loss of peaks from superposition (overlap) with neighboring peaks to the loss arising from lying below the limit of detection (LOD) imposed by a finite signal-to-noise ratio (SNR). The simulations are made using amplitude pdfs based on the Enke-Nagels data as functions of relative column efficiency, i.e., saturation, and SNR. Results show that at the highest efficiencies, the lowest-amplitude peaks are lost below the LOD. However, at small and medium efficiencies, peak overlap is the dominant loss mechanism, suggesting that low-level components will not be found easily in liquid chromatography with single channel detectors regardless of SNR. A simple treatment shows that a multichannel detector, e.g., a mass spectrometer, is necessary to expose more low-level components.

Measuring porosities of chromatographic columns utilizing a mass-based total pore-blocking method: Superficially porous particles and pore-blocking critical pressure mechanism.

Experimentally determined total, interstitial and intraparticle porosity values are necessary to equate theory, simulation and experimental column performance. This paper reports a study of a mass-based technique for determining total, interstitial and intraparticle porosity measurements based on the total pore-blocking (TPB) method. Commercially available superficially porous particle (SPP) columns, in a variety of small-pore and wide-pore materials, with both hydrophobic and hydrophilic surfaces, are utilized as samples. The results are compared with previously determined literature values for a number of columns and contrasted with HPLC-based elution methods. This method uses only a high-precision balance and an HPLC pump. A simple theoretical analysis of the TPB method using the Young-Laplace equation shows the pressure bounds and flow rate constraints of the method which ensure pore blocking stability. The results suggest that particles with small-pore diameters can be analyzed over a range of solvent clearing pressures and flow rates. However, wide-pore materials, typically with pore diameters in excess of 400 Å, have very low critical pressures and are difficult to determine without losing the pore blocking component. Small mass differences between clearing solvents are shown to present a challenge for measuring the interstitial volume.

Column selection for comprehensive two-dimensional liquid chromatography using the hydrophobic subtraction model.

Two-dimensional (2D) liquid chromatography (2DLC) methods have grown in popularity due to their enhanced peak capacity that allows for resolving complex samples. Given the large number of commercially available column types, one of the major challenges in implementing 2DLC methods is the selection of suitable column pairs. Column selection is typically informed by chemical intuition with subsequent experimental optimization. In this work a computational screening method for 2DLC is proposed whereby virtual 2D chromatograms are calculated utilizing the Snyder-Dolan hydrophobic subtraction model (HSM) for reversed-phase column selectivity. Towards this end, 319 225 column pairs resulting from the combination of 565 columns and 100 sets of 1000 diverse analytes are examined. Compared to other screening approaches, the present method is highly predictive for column pairs that are able to resolve the largest number of analytes. This approach shows a strong sensitivity to the choice of the second dimension column (having a shorter operating time) and a preference for those with embedded polar moieties, whereas a relatively weak preference for C18 and phenyl columns is found for the first dimension.

Simulation and theory of open-tube dispersion in short and long capillaries with slip boundaries and retention.

Using random walk techniques, high resolution simulations of zone shape are conducted in open capillary tubes for short and long tube conditions. Finite size solutes are used as tracers in this treatment. Slip flow boundary conditions and wall retention are utilized as needed. These simulations are able to reproduce previous work in short and long tubes. For the short tube case where dispersion does not asymptotically approach the classic Taylor-Aris and Golay solutions, the effect of slip flow boundaries in the transient region shows zone shapes with abbreviated tails where the larger slip flow values cause zone compression. The use of slip flow to lower dispersion in capillary-based, wall-coated separations is shown to favor long tube behavior. This is because slip flow is relevant for cases where slip lengths are fractions of small capillary tube diameters. Incorporating slip flow into transport in capillaries favors a very small capillary radius where the cross-sectional diffusion length is very small and sampling times are fast. The purely convective zone shape with slip flow boundaries is derived analytically. Applications for this type of separation, guided by both analytical theory and simulation, show the potential for nano-sized capillary tubes less than 1 µm in diameter and favor very fast isocratic separations. Using long tube retention theory with slip boundaries shows that the dispersion-reducing region is most important in the range 0 ≤ k' ≤ 1, a relatively small retention window. Further discussion of the gradient elution technique and dispersion in packed beds suggests that the general usage of slip flow boundaries is restricted in liquid phase separation systems.

Ellipsoidal particles for liquid chromatography: Fluid mechanics, efficiency and wall effects.

Ellipsoidal particles are investigated as packing media for liquid chromatography using high resolution fluid mechanics and Brownian dynamics simulations. The simulations are conducted with packed capillary columns, as well as beds with periodic boundary conditions (PBCs) to study transport in the absence of wall effects. The performance of ellipsoidal particles is evaluated over a range of aspect ratios. The definition of effective diameter used to compare sphere and ellipsoidal particle performance metrics is presented and discussed along with scaling relationships which are necessary to compare sphere and ellipsoidal particle packs. Ellipsoidal particle packs are found to be inferior to sphere packs using PBCs to study chromatographic dispersion. The separation impedance was calculated with PBCs and shown to be approximately the same with ellipsoidal particles as those of spheres. Efficiency of ellipsoidal packs, as measured by plate height, is lower than spherical particle packs and the pressure drop is higher than sphere packs when using PBCs. However, a smaller wall effect is shown for ellipsoidal particles when packing cylindrical capillaries. Radial variations in packing porosity and in flow within the wall region are smaller for ellipsoidal packings. The minimum reduced plate height and the separation impedance for the packed capillaries clearly demonstrate the advantages of ellipsoidal particles compared to spherical particles. This predicted performance advantage remains to be demonstrated in actual practice.

Using molecular simulations to probe pore structures and polymer partitioning in size exclusion chromatography.

Molecular simulations have been extensively utilized to understand and predict the polymer partitioning in size-exclusion chromatography (SEC). However, idealized pore models (e.g., cylindrical, spherical, and slit pores) were often used to represent the porous media in an SEC column, which leads to significant deviations in describing the geometry and the size of the pores. In this work, several complex pore models were derived from body-centered cubic, random, and gel packing of monodisperse spherical sol particles using simulation methodology. The mechanical stabilities of these structures were determined based on particle coordination numbers. Pore size distributions of these porous structures were compared to a commercially available, wide-pore superficially porous particle. Then, Gibbs ensemble Monte Carlo simulations were performed to compute the pore-to-bulk partitioning coefficient KSEC of a polymer chain with complex pore models. The effects of particle size, packing structure, and porosity on KSEC were explored. In addition, structural analysis provides insight into the conformation of polymers in the pores and its effect on the partitioning behavior. This study promotes the understanding of pore structures in SEC columns and enables more accurate predictions of KSEC with less ambiguity in pore geometry.

Transport properties and size exclusion effects in wide-pore superficially porous particles.

The effects of hydrodynamic radius on the transport of solute molecules in packed beds of wide-pore superficially porous particles (SPP) are studied using pore-scale simulation. The free molecular diffusion rate varies with radius through the Stokes-Einstein relation. Lattice Boltzmann and Langevin methods are used to model fluid motion and the transport of an ensemble of solute molecules in the fluid, providing statistics on solute concentration, flux, molecule age and residence time, as a function of depth in the SPP. Intraparticle effective diffusion and bed dispersion coefficients are calculated and correlated with the hydrodynamic radius and accessible porosity. The relative importance of convection and diffusion are found to depend on the molecule (tracer) size through the diffusion rate, and convection effects are more significant for larger, slower-diffusing molecules. When larger molecules are utilized, the intraparticle concentration is reduced in proportion to the local particle porosity, leading to a natural definition of the accessible porosity used in size exclusion chromatography (SEC). Although the pore shape is complex, the SEC constant K can be calculated directly from simulation. Simulation demonstrates that the effective diffusion coefficient is elevated near the particle hull, which is largely open to interstitial flow, and decreases with depth into the particle. All molecules studied here have transport access to the entire particle depth, although the accessible volume at a given depth depends on their size. The first passage time into the particle is well predicted by the diffusion rate, but residence time is influenced by convection, shortening the average visit duration. These results are of interest in "perfusion" chromatography where convection is thought to increase separation efficiency for large biomolecules.

Orthogonality measurements for multidimensional chromatography in three and higher dimensional separations.

Orthogonality metrics (OMs) for three and higher dimensional separations are proposed as extensions of previously developed OMs, which were used to evaluate the zone utilization of two-dimensional (2D) separations. These OMs include correlation coefficients, dimensionality, information theory metrics and convex-hull metrics. In a number of these cases, lower dimensional subspace metrics exist and can be readily calculated. The metrics are used to interpret previously generated experimental data. The experimental datasets are derived from Gilar's peptide data, now modified to be three dimensional (3D), and a comprehensive 3D chromatogram from Moore and Jorgenson. The Moore and Jorgenson chromatogram, which has 25 identifiable 3D volume elements or peaks, displayed good orthogonality values over all dimensions. However, OMs based on discretization of the 3D space changed substantially with changes in binning parameters. This example highlights the importance in higher dimensions of having an abundant number of retention times as data points, especially for methods that use discretization. The Gilar data, which in a previous study produced 21 2D datasets by the pairing of 7 one-dimensional separations, was reinterpreted to produce 35 3D datasets. These datasets show a number of interesting properties, one of which is that geometric and harmonic means of lower dimensional subspace (i.e., 2D) OMs correlate well with the higher dimensional (i.e., 3D) OMs. The space utilization of the Gilar 3D datasets was ranked using OMs, with the retention times of the datasets having the largest and smallest OMs presented as graphs. A discussion concerning the orthogonality of higher dimensional techniques is given with emphasis on molecular diversity in chromatographic separations. In the information theory work, an inconsistency is found in previous studies of orthogonality using the 2D metric often identified as %O. A new choice of metric is proposed, extended to higher dimensions, characterized by mixes of ordered and random retention times, and applied to the experimental datasets. In 2D, the new metric always equals or exceeds the original one. However, results from both the original and new methods are given.

Superficially porous particles with 1000Å pores for large biomolecule high performance liquid chromatography and polymer size exclusion chromatography.

To facilitate mass transport and column efficiency, solutes must have free access to particle pores to facilitate interactions with the stationary phase. To ensure this feature, particles should be used for HPLC separations which have pores sufficiently large to accommodate the solute without restricted diffusion. This paper describes the design and properties of superficially porous (also called Fused-Core®, core shell or porous shell) particles with very large (1000Å) pores specifically developed for separating very large biomolecules and polymers. Separations of DNA fragments, monoclonal antibodies, large proteins and large polystyrene standards are used to illustrate the utility of these particles for efficient, high-resolution applications.

Size exclusion chromatography with superficially porous particles.

A comparison is made using size-exclusion chromatography (SEC) of synthetic polymers between fully porous particles (FPPs) and superficially porous particles (SPPs) with similar particle diameters, pore sizes and equal flow rates. Polystyrene molecular weight standards with a mobile phase of tetrahydrofuran are utilized for all measurements conducted with standard HPLC equipment. Although it is traditionally thought that larger pore volume is thermodynamically advantageous in SEC for better separations, SPPs have kinetic advantages and these will be shown to compensate for the loss in pore volume compared to FPPs. The comparison metrics include the elution range (smaller with SPPs), the plate count (larger for SPPs), the rate production of theoretical plates (larger for SPPs) and the specific resolution (larger with FPPs). Advantages to using SPPs for SEC are discussed such that similar separations can be conducted faster using SPPs. SEC using SPPs offers similar peak capacities to that using FPPs but with faster operation. This also suggests that SEC conducted in the second dimension of a two-dimensional liquid chromatograph may benefit with reduced run time and with equivalently reduced peak width making SPPs advantageous for sampling the first dimension by the second dimension separator. Additional advantages are discussed for biomolecules along with a discussion of optimization criteria for size-based separations.

Orthogonal separations: Comparison of orthogonality metrics by statistical analysis.

Twenty orthogonality metrics (OMs) derived from convex hull, information theory, fractal dimension, correlation coefficients, nearest neighbor distances and bin-density techniques were calculated from a diverse group of 47 experimental two-dimensional (2D) chromatograms. These chromatograms comprise two datasets; one dataset is a collection of 2D chromatograms from Peter Carr's laboratory at the University of Minnesota, and the other dataset is based on pairs of one-dimensional chromatograms previously published by Martin Gilar and coworkers (Waters Corp.). The chromatograms were pooled to make a third or combined dataset. Cross-correlation results suggest that specific OMs are correlated within families of nearest neighbor methods, correlation coefficients and the information theory methods. Principal component analysis of the OMs show that none of the OMs stands out as clearly better at explaining the data variance than any another OM. Principal component analysis of individual chromatograms shows that different OMs favor certain chromatograms. The chromatograms exhibit a range of quality, as subjectively graded by nine experts experienced in 2D chromatography. The subjective (grading) evaluations were taken at two intervals per expert and demonstrated excellent consistency for each expert. Excellent agreement for both very good and very bad chromatograms was seen across the range of experts. However, evaluation uncertainty increased for chromatograms that were judged as average to mediocre. The grades were converted to numbers (percentages) for numerical computations. The percentages were correlated with OMs to establish good OMs for evaluating the quality of 2D chromatograms. Certain metrics correlate better than others. However, these results are not consistent across all chromatograms examined. Most of the nearest neighbor methods were observed to correlate poorly with the percentages. However, one method, devised by Clark and Evans, appeared to work moderately well. Products of OMs show better correlation with the percentages than do single OMs. Product OMs that utilize one discretized metric paired with the convex hull relative area, which measures overall zone occupancy, perform well in determining the "best" chromatogram among both datasets and the combined dataset. A definition of chromatographic orthogonality is suggested that is based on maximizing the values of OMs or OM products. This optimization criterion suggests using the product of a global metric that measures the utilization of separation space (e.g., the convex hull relative area) and a local metric that measures peak spacing (e.g., the box-counting fractal dimension). The "best" column pairs for 2D chromatography are chosen by the product of these OMs.

Molecular simulation studies of reversed-phase liquid chromatography.

Over the past 20 years, molecular simulation methods have been applied to the modeling of reversed-phase liquid chromatography (RPLC). The purpose of these simulations was to provide a molecular-level understanding of: (i) the structure and dynamics of the bonded phase and its interface with the mobile phase, (ii) the interactions of analytes with the bonded phase, and (iii) the retention mechanism for different analytes. However, the investigation of chromatographic systems poses significant challenges for simulations with respect to the accuracy of the molecular mechanics force fields and the efficiency of the sampling algorithms. This review discusses a number of aspects concerning molecular simulation studies of RPLC systems including the historical development of the subject, the background needed to understand the two prevalent techniques, molecular dynamics (MD) and Monte Carlo (MC) methods, and the wealth of insight provided by these simulations. Examples from the literature employing MD approaches and from the authors' laboratory using MC methods are discussed. The former can provide information on chain dynamics and transport properties, whereas the latter techniques are uniquely suited for the investigation of phase and sorption equilibria that underly RPLC retention, and both can be used to elucidate the bonded-chain conformations and solvent distributions.

Comparison of orthogonality estimation methods for the two-dimensional separations of peptides.

In two-dimensional chromatography, the orthogonality of separation is important for achieving high peak capacity. In this paper, a number of different metrics are compared as measures of orthogonality. Six peptide elution data sets acquired on different stationary phases are plotted against reversed phase retention data and examined as two-dimensional chromatographic pairs. The data, including six in silico prepared data pairs, are utilized to challenge and compare selected orthogonality metrics. The metrics include correlation coefficients, mutual information, box-counting dimensionality, and surface fractional coverage with different hulls. Although correlation coefficients were found to be less suited for the intended purpose, other methods can provide a suitable measure of orthogonality. The presented results are discussed in terms of method utility, simplicity, and applicability for statistically small sets of chromatographic data. Two of the methods, box counting dimensionality and fractional coverage, were found to be mathematically related.

Modeling of dispersion in a polymeric chromatographic monolith.

Dispersion in a commercial polymeric monolith was simulated on a sample geometry obtained by direct imaging using high-resolution electron microscopy. A parallelized random walk algorithm, implemented using a velocity field obtained previously by the lattice-Boltzmann method, was used to model mass transfer. Both point particles and probes of finite size were studied. Dispersion simulations with point particles using periodic boundaries resulted in plate heights that varied almost linearly with flow rate, at odds with the weaker dependence suggested by experimental observations and predicted by theory. This discrepancy resulted from the combined effect of the artificial symmetry in the velocity field and the periodic boundaries implemented to emulate macroscopic column lengths. Eliminating periodicity and simulating a single block length instead resulted in a functional dependence of plate heights on flow rate more in accord with experimental trends and theoretical predictions for random media. The lower values of the simulated plate heights than experimental ones are attributed in part to the presence of walls in real systems, an effect not modeled by the algorithm. On the other hand, analysis of transient dispersion coefficients and comparison of lateral particle positions at the entry and exit hinted at non-asymptotic behavior and a strong degree of correlation that was presumably a consequence of preferential high-velocity pathways in the raw sample block. Simulations with finite-sized probes resulted in particle trajectories that frequently terminated at narrow constrictions of the geometry. The amount of entrapment was predicted to increase monotonically with flow rate, evidently due to the relative contributions to transport by convection that carries particles to choke-points and diffusion that dislodges these entrapped particles. The overall effect is very similar to a flow-dependent entrapment phenomenon previously observed experimentally for adenovirus.

A molecular simulation study of the effects of stationary phase and solute chain length in reversed-phase liquid chromatography.

The effects of stationary phase and solute chain length are probed by carrying out Monte Carlo simulations of dimethyl triacontyl (C30), dimethyl octadecyl (C18), dimethyl octyl (C8), and trimethyl (C1) silane grafted, and bare silica stationary phases in contact with a water/methanol mobile phase and by examining the retention of solutes from 1 to 14 carbons in length. Fairly small differences in structure are observed when comparing the C30, C18, C8 systems and the retention mechanism of nonpolar alkane solutes shows contribution from both partitioning and adsorption on all three of these stationary phases. Unlike in the other systems, the mobile phase solvent is highly structured at its interface with the C1 and bare silica phases, the former being enriched in methanol and the latter in water. Alkane solutes are unretained at the bare silica surface while alcohol solutes are only slightly enriched at the silica surface due to hydrogen bonding with surface silanols and surface bound solvent. With regard to solute size, it appears that the retention mechanism is not affected by the chain length of the solute.

Molecular simulations of retention in chromatographic systems: use of biased Monte Carlo techniques to access multiple time and length scales.

The use of configurational-bias Monte Carlo simulations in the Gibbs ensemble allows for the sampling of phenomena that occur on vastly different time and length scales. In this review, applications of this simulation approach to probe retention in gas and reversed-phase liquid chromatographic systems are discussed. These simulations provide an unprecedented view of the retention processes at the molecular-level and show excellent agreement with experimental retention data.

The statistical overlap theory of chromatography using power law (fractal) statistics.

The chromatographic dimensionality was recently proposed as a measure of retention time spacing based on a power law (fractal) distribution. Using this model, a statistical overlap theory (SOT) for chromatographic peaks is developed that estimates the number of peak maxima as a function of the chromatographic dimension, saturation and scale. Power law models exhibit a threshold region whereby below a critical saturation value no loss of peak maxima due to peak fusion occurs as saturation increases. At moderate saturation, behavior is similar to the random (Poisson) peak model. At still higher saturation, the power law model shows loss of peaks nearly independent of the scale and dimension of the model. The physicochemical meaning of the power law scale parameter is discussed and shown to be equal to the Boltzmann-weighted free energy of transfer over the scale limits. The scale is discussed. Small scale range (small ß) is shown to generate more uniform chromatograms. Large scale range chromatograms (large ß) are shown to give occasional large excursions of retention times; this is a property of power laws where "wild" behavior is noted to occasionally occur. Both cases are shown to be useful depending on the chromatographic saturation. A scale-invariant model of the SOT shows very simple relationships between the fraction of peak maxima and the saturation, peak width and number of theoretical plates. These equations provide much insight into separations which follow power law statistics.

Retention mechanism for polycyclic aromatic hydrocarbons in reversed-phase liquid chromatography with monomeric stationary phases.

Reversed-phase liquid chromatography (RPLC) is the foremost technique for the separation of analytes that have very similar chemical functionalities, but differ only in their molecular shape. This ability is crucial in the analysis of various mixtures with environmental and biological importance including polycyclic aromatic hydrocarbons (PAHs) and steroids. A large amount of effort has been devoted to studying this phenomenon experimentally, but a detailed molecular-level description remains lacking. To provide some insight on the mechanism of shape selectivity in RPLC, particle-based simulations were carried out for stationary phases and chromatographic parameters that closely mimic those in an experimental study by Sentell and Dorsey [J. Chromatogr. 461 (1989) 193]. The retention of aromatic hydrocarbons ranging in size from benzene to the isomeric PAHs of the formula C(18)H(12) was examined for model RPLC systems consisting of monomeric dimethyl octadecylsilane (ODS) stationary phases with surface coverages ranging from 1.6 to 4.2 µmol/m(2) (i.e., stationary phases yielding low to intermediate shape selectivity) in contact with a 67/33 mol% acetonitrile/water mobile phase. The simulations show that the stationary phase acts as a very heterogeneous environment where analytes with different shapes prefer different spatial regions with specific local bonding environments of the ODS chains. However, these favorable retentive regions cannot be described as pre-existing cavities because the chain conformation in these local stationary phase regions adapts to accommodate the analytes.

Modeling of flow in a polymeric chromatographic monolith.

The flow behavior of a commercial polymeric monolith was investigated by direct numerical simulations employing the lattice-Boltzmann (LB) methodology. An explicit structural representation of the monolith was obtained by serial sectioning of a portion of the monolith and imaging by scanning electron microscopy. After image processing, the three-dimensional structure of a sample block with dimensions of 17.8 µm × 17.8 µm × 14.1 µm was obtained, with uniform 18.5 nm voxel size. Flow was simulated on this reconstructed block using the LB method to obtain the velocity distribution, and in turn macroscopic flow properties such as the permeability and the average velocity. The computed axial velocity distribution exhibits a sharp peak with an exponentially decaying tail. Analysis of the local components of the flow field suggests that flow is not evenly distributed throughout the sample geometry, as is also seen in geometries that exhibit preferential flow paths, such as sphere pack arrays with defects. A significant fraction of negative axial velocities are observed; the largest of these are due to flow along horizontal pores that are also slightly oriented in the negative axial direction. Possible implications for mass transfer are discussed.