Nonlinear Retention Relationships in Reversed-Phase Chromatography

Numerical Approximation of a Nonequilibrium Model of Gradient Elution Chromatography Considering Different Functional Relationships between Model Parameters and Solvent Composition

In this paper, a rigorous theoretical study is conducted to analyze the influence of varying solvent compositions on the retention characteristics of elution profiles within a fixed-bed liquid chromatographic column. In gradient chromatography, the propagation speed of elution profiles is manipulated through a progressive variation in the mobile-phase composition. Consequently, enhanced separation of the mixture components can be achieved together with a reduction in the requisite recycling times for subsequent injections. In other words, both the efficiency and the selectivity of the column can be enhanced. The lumped kinetic model coupled with the convection-diffusion equation for the volume fraction of the solvent is applied to simulate the process. The resulting nonlinear model equations are numerically solved by applying a semidiscrete second-order finite-volume method. The numerical solutions are utilized to quantify the effects of gradient starting and ending times, solvent composition, solvent strength parameters, and gradient slope on the concentration profiles. Additionally, temporal numerical moments are plotted versus the starting and ending times of the gradient, and standard performance criteria are presented for evaluating the process performance. The outcomes of this investigation will contribute to further enhancements in gradient elution chromatography.

The Solvation Shell of Small Solutes in Aqueous-Organic Solvent Mixtures and Its Implications for Reversed-Phase Liquid Chromatography

Reversed-phase liquid chromatography (RPLC) operates with water-organic solvent (W-OS) mobile phases where preferential solvation (PS) of solutes is likely. To investigate the relevance of the solute solvation shell in the mobile phase for RPLC retention, we combine data from molecular dynamics simulations of small, neutral solutes (six analytes and two dead time markers) in W-methanol (MeOH) and W-acetonitrile (ACN) mixtures with corresponding retention data obtained on an RPLC column over a wide range of W/OS ratios. Data derived from Kirkwood-Buff integrals show PS by the OS for analytes vs low or negative PS for dead time markers. W-ACN mixtures generate a higher amount of PS than W-MeOH mixtures, which contributes to the higher eluent strength of ACN in RPLC. Difference spatial distribution functions reveal anisotropic solvation shells with OS excess at hydrocarbon elements and W excess at functional groups, predicting that retention by the hydrophobic stationary phase is favored by hydrocarbon elements and limited by functional groups. Analysis of solute-solvent hydrogen bonds pinpoints the hydrogen-bond requirements toward W as the retention-limiting factor. The relation between the solute solvation shell and retention confirms the importance of W-OS and solute-W hydrogen bonding for RPLC retention.

Computer-driven optimization of complex gradients in comprehensive two-dimensional liquid chromatography

Method development in comprehensive two-dimensional liquid chromatography (LC × LC) is a complicated endeavor. The dependency between the two dimensions and the possibility of incorporating complex gradient profiles, such as multi-segmented gradients or shifting gradients, renders method development by "trial-and-error" time-consuming and highly dependent on user experience. In this work, an open-source algorithm for the automated and interpretive method development of complex gradients in LC × LC-mass spectrometry (MS) was developed. A workflow was designed to operate within a closed-loop that allowed direct interaction between the LC × LC-MS system and a data-processing computer which ran in an unsupervised and automated fashion. Obtaining accurate retention models in LC × LC is difficult due to the challenges associated with the exact determination of retention times, curve fitting because of the use of gradient elution, and gradient deformation. Thus, retention models were compared in terms of repeatability of determination. Additionally, the design of shifting gradients in the second dimension and the prediction of peak widths were investigated. The algorithm was tested on separations of a tryptic digest of a monoclonal antibody using an objective function that included the sum of resolutions and analysis time as quality descriptors. The algorithm was able to improve the separation relative to a generic starting method using these complex gradient profiles after only four method-development iterations (i.e., sets of chromatographic conditions). Further iterations improved retention time and peak width predictions and thus the accuracy in the separations predicted by the algorithm.

Retention modeling and adsorption mechanisms in reversed-phase liquid chromatography

To interpret the dependence of solute retention behavior on modifier content in reversed-phase liquid chromatography, a theoretical framework, based on the concentration dependence of solvophobic forces imposed on solutes and the competitive adsorptions of solutes and solvent modifiers, was proposed. The generality of the developed model was demonstrated by comparing the model with conventional retention models. The linear dependence of the Gibbs energy change of solute adsorption with respect to the modifier concentration was assumed, and the model was fitted to the experimental results, with good agreement demonstrated between the experimental data and the model. Retention behaviors were inferred to be determined by two key dimensionless groups that represented the reductions in the retention factors resulting from a weakened solvophobic interaction and modifier competitive adsorption. The retention behaviors were successfully deconvoluted for each contribution as a function of the modifier concentration by using the fitted parameters. The effects of both contributions on the retention behaviors were enhanced for the solutes with aromatic groups. The standard Gibbs energy change _SL^o of benzene adsorption was found to depend linearly on the number of modifier molecules present but independent of modifier identity. For the solutes associated with hydrogen-bonding groups, the degree of reduction in the solvophobic interactions was considerably reduced. Hence, the relative contributions of both mechanisms to solute retention depend greatly on the solute structure. Perturbation method was performed to investigate the modifier adsorption mechanisms. The results show that the standard Gibbs energy change _SL^o for the first-layer adsorption of modifiers changed linearly with the carbon number of modifier molecule. These results demonstrated that the proposed model can offer a physically consistent quantitative description of retention when solvent composition is varied.

A new strategy for the computer-assisted development of reversed-phase liquid chromatography separation methods of unknown sample mixtures

A new strategy for the computer-assisted methods development in the reversed-phase liquid chromatographic separations of unknown sample mixtures has been developed using the latent spectral information in chromatogram raw data files of appropriately designed experiments, rather than resorting to elemental information functions (e.g., the number of peaks in chromatograms or similar criteria). The strategy developed allows unification of the approach for samples of both known and unknown composition and, thus, provide a general strategy for computer-aided tools in the chromatography laboratory. The operation principle of this strategy departs from extracting the spectra of components in the mixture chromatograms by resorting to multivariate curve resolution-alternating least squares (MCR-ALS). This technique allows the estimation of the true spectra for the individual components except when they have identical spectra or are fully overlapped. Thus, a convenient experimental design will try to perform separations of the sample mixture having at least partial resolution of components in some runs. This will allow estimating the spectra of components and, then, assign these components to the peaks in each run chromatogram. In this way, a retention model can be built for each component so computerized optimization process can be developed to provide the chromatographer with the best possible separation programs. Following this approach, strategies for sample mixtures of known and unknown composition are only different in the need of an initial spectrum discovery process for unknown mixtures and therefore a real general approach for the computer-assisted LC methods development is now available for the first time.

Rebirth of recycling liquid chromatography with modern chromatographic columns : Extension to gradient elution

Twin column recycling semi-preparative liquid chromatography (TCRLC) is revived to prepare small amount (∼ 1 mg) of a pure targeted compound, which cannot be isolated by conventional preparative liquid chromatography. In this work, TCRLC is extended to gradient elution. The first step of this modified process consists of a gradient step, which eliminates both early and late impurities. If not discarded, some late impurities could echo during the second isocratic recycling step of the process and compromise the purity level required for the targeted compound. Additionally, the entire gradient TCRLC (GTCRLC) process is automated regarding the eluent composition programmed and the actuation times of two valves: one two-position four-port divert valve enables to shave the targeted compound from early and late impurities during the initial gradient step. The second two-position six-port recycling valve ensures the complete baseline resolution between the band of the targeted compound and those of the closest impurities, which are not fully eliminated after the initial gradient step. The automation of the whole GTCRLC process is achieved by running four preliminary scouting gradient runs (at four different relative gradient times, t_gt₀= 2, 6, 18, and 54, where t₀ is the hold-up column time) for the accurate determination of the thermodynamics (lnk versus φ plots of the retention factor as a function of the mobile phase composition) of the first impurity, the targeted compound(s), and of the last impurity. The automated GTCRLC process was successfully applied for the isolation of a polycyclic aromatic hydrocarbon (PAH), chrysene, from a complex mixture of PAHs containing two nearly co-eluting impurities (benzo[a]anthracene and triphenylene) and nine other early/late impurities (sample volume injected: 1 mL, 7.8 mm × 150 mm Sunfire-C₁₈ column, acetonitrile/water eluent mixtures, T= 55 ^∘C, 20 cycles, baseline separation in less than two hours). Additionally, the GTCRLC process is advantageously used to isolate and baseline separate the vitamins D₂ and D₃ initially present in a milk extract mixture (0.3 mL sample injection volume, 7.8 mm × 150 mm Sunfire-C₁₈ column, methanol/water eluent mixtures, T= 65 ^∘C, 14 cycles needed in 1.5 hours). These results open promising avenues toward an effective preparation of unknown targeted compounds before further physico-chemical characterization and unambiguous identification.

Perspective on the Future Approaches to Predict Retention in Liquid Chromatography

The demand for rapid column screening, computer-assisted method development and method transfer, and unambiguous compound identification by LC/MS analyses has pushed analysts to adopt experimental protocols and software for the accurate prediction of the retention time in liquid chromatography (LC). This Perspective discusses the classical approaches used to predict retention times in LC over the last three decades and proposes future requirements to increase their accuracy. First, inverse methods for retention prediction are essentially applied during screening and gradient method optimization: a minimum number of experiments or design of experiments (DoE) is run to train and calibrate a model (either purely statistical or based on the principles and fundamentals of liquid chromatography) by a mere fitting process. They do not require the accurate knowledge of the true column hold-up volume V₀, system dwell volume V_dwell (in gradient elution), and the retention behavior (k versus the content of strong solvent φ, temperature T, pH, and ionic strength I) of the analytes. Their relative accuracy is often excellent below a few percent. Statistical methods are expected to be the most attractive to handle very complex retention behavior such as in mixed-mode chromatography (MMC). Fundamentally correct retention models accounting for the simultaneous impact of φ, I, pH, and T in MMC are needed for method development based on chromatography principles. Second, direct methods for retention prediction are ideally suited for accurate method transfer from one column/system configuration to another: these quality by design (QbD) methods are based on the fundamentals and principles of solid-liquid adsorption and gradient chromatography. No model calibration is necessary; however, they require universal conventions for the accurate determination of true retention factors (for 1 < k < 30) as a function of the experimental variables (φ, T, pH, and I) and of the true column/system parameters (V₀, V_dwell, dispersion volume, σ, and relaxation volume, τ, of the programmed gradient profile at the column inlet and gradient distortion at the column outlet). Finally, when the molecular structure of the analytes is either known or assumed, retention prediction has essentially been made on the basis of statistical approaches such as the linear solvation energy relationships (LSERs) and the quantitative structure retention relationships (QSRRs): their ability to accurately predict the retention remains limited within 10-30%. They have been combined with molecular similarity approaches (where the retention model is calibrated with compounds having structures similar to that of the targeted analytes) and artificial intelligence algorithms to further improve their accuracy below 10%. In this Perspective, it is proposed to adopt a more rigorous and fundamental approach by considering the very details of the solid-liquid adsorption process: Monte Carlo (MC) or molecular dynamics (MD) simulations are promising tools to explain and interpret retention data that are too complex to be described by either empirical or statistical retention models.

Detailed numerical study of the peak shapes of neutral analytes injected at high solvent strength in short reversed-phase liquid chromatography columns and comparison with experimental observations

We report on a numerical investigation of the different steps in the development of the spatial concentration profiles developing along the axis of a liquid chromatography column when injecting large relative volumes (>10 to 20% of column volume) of analytes dissolved in a high solvent strength solvent band as can be encountered in the second dimension (²D) column of a two-dimensional liquid chromatography (2D-LC) system. More specifically, we made a detailed study of the different retention and the axial band broadening effects leading to the double-headed peak shapes or strongly fronting peaks that can be experimentally observed under certain conditions in 2D-LC. The establishment of these intricate peak profiles is discussed in all its fine, mechanistic details. The effect of the volume of the column, the volume and the shape of the sample band, the retention properties of the analyte and the band broadening experienced by the analytes and the sample solvent are investigated. A good agreement between the simulations and the experimental observations with caffeine and methylparaben injected in acetonitrile/water (ACN/H₂O) mobile phase with different injection volumes is obtained. Save the difference in dwell volume, key features of experimental and simulated chromatograms agree within a few %. The simulations are also validated against a number of simple mathematical rules of thumb that can be established to predict the occurrence of a breakthrough fraction and estimate the amount of breakthrough.

Deep Q-learning for the selection of optimal isocratic scouting runs in liquid chromatography

An important challenge in chromatography is the development of adequate separation methods. Accurate retention models can significantly simplify and expedite the development of adequate separation methods for complex mixtures. The purpose of this study was to introduce reinforcement learning to chromatographic method development, by training a double deep Q-learning algorithm to select optimal isocratic scouting runs to generate accurate retention models. These scouting runs were fit to the Neue-Kuss retention model, which was then used to predict retention factors both under isocratic and gradient conditions. The quality of these predictions was compared to experimental data points, by computing a mean relative percentage error (MRPE) between the predicted and actual retention factors. By providing the reinforcement learning algorithm with a reward whenever the scouting runs led to accurate retention models and a penalty when the analysis time of a selected scouting run was too high (> 1h); it was hypothesized that the reinforcement learning algorithm should by time learn to select good scouting runs for compounds displaying a variety of characteristics. The reinforcement learning algorithm developed in this work was first trained on simulated data, and then evaluated on experimental data for 57 small molecules - each run at 10 different fractions of organic modifier (0.05 to 0.90) and four different linear gradients. The results showed that the MRPE of these retention models (3.77% for isocratic runs and 1.93% for gradient runs), mostly obtained via 3 isocratic scouting runs for each compound, were comparable in performance to retention models obtained by fitting the Neue-Kuss model to all (10) available isocratic datapoints (3.26% for isocratic runs and 4.97% for gradient runs) and retention models obtained via a "chromatographer's selection" of three scouting runs (3.86% for isocratic runs and 6.66% for gradient runs). It was therefore concluded that the reinforcement learning algorithm learned to select optimal scouting runs for retention modeling, by selecting 3 (out of 10) isocratic scouting runs per compound, that were informative enough to successfully capture the retention behavior of each compound.

Recent applications of chemometrics in one- and two-dimensional chromatography

The proliferation of increasingly more sophisticated analytical separation systems, often incorporating increasingly more powerful detection techniques, such as high-resolution mass spectrometry, causes an urgent need for highly efficient data-analysis and optimization strategies. This is especially true for comprehensive two-dimensional chromatography applied to the separation of very complex samples. In this contribution, the requirement for chemometric tools is explained and the latest developments in approaches for (pre-)processing and analyzing data arising from one- and two-dimensional chromatography systems are reviewed. The final part of this review focuses on the application of chemometrics for method development and optimization.

Evaluation of three temperature- and mobile phase-dependent retention models for reversed-phase liquid chromatographic retention and apparent retention enthalpy

Predicting retention and enthalpy allows for the simulation and optimization of advanced chromatographic techniques including gradient separations, temperature-assisted solute focusing, multidimensional liquid chromatography, and solvent focusing. In this paper we explore the fits of three expressions for retention as a function of mobile phase composition and temperature to retention data of 101 small molecules in reversed phase liquid chromatography. The three retention equations investigated are those by Neue and Kuss (NK) and two different equations by Pappa-Louisi et al., one based on a partition model (PL-P) and one based on an adsorption model (PL-A). More than 25 000 retention factors were determined for 101 small molecules under various mobile phase and temperature conditions. The pure experimental uncertainty is very small, approximately 0.22% uncertainty in retention factors measured on the same day (2.1% when performed on different days). Each of the three equations for ln(k) was fit to the experimental data based on a least-squares approach and the results were analyzed using lack-of-fit residuals. The PL-A model, while complex, gives the best overall fits. In addition to examining the equations' adequacy for retention, we also examined their use for apparent retention enthalpy. This enthalpy can be predicted by taking the derivative of these expressions with respect to the inverse of absolute temperature. The numerical values of the fitted parameters based on retention data can then be used to predict retention enthalpy. These enthalpy predictions were compared to those obtained from a modified van 't Hoff equation that included a quadratic term in inverse temperature. Based on analysis of 1 211 van 't Hoff plots (solute-mobile phase-day combinations), ninety-eight percent showed a significantly better fit when using the modified van 't Hoff expression, justifying its use to provide apparent enthalpies as a function of mobile phase composition and temperature. The foregoing apparent enthalpies were compared to the apparent enthalpies predicted by the three models. The PL-A model, which contains a temperature dependent enthalpy, provided the best enthalpy prediction. However, there is virtually no correlation between the overall lack of fit to experimental ln(k) for each model and the corresponding lack of fit of the linear (in 1/T) van 't Hoff expression. Thus, the temperature-dependent enthalpy is apparently not the cause of a model's ability to fit ln(k) as a function of mobile phase composition and temperature. The value in these expressions is their ability to predict chromatograms, allowing for optimization of an advanced chromatographic technique. The two simpler models NK and PL-P, which do not contain a temperature dependent enthalpy, have their merits in modelling retention (NK being the better of the two) and enthalpy (PL-P being the better of the two) if a simpler expression is required for a given application.

Retention and effective diffusion of model metabolites on porous graphitic carbon

The study of metabolites in biological samples is of high interest for a wide range of biological and pharmaceutical applications. Reversed phase liquid chromatography is a common technique used for the separation of metabolites, but it provides little retention for polar metabolites. An alternative to C18 bonded phases, porous graphitic carbon has the ability to provide significant retention for both non-polar and polar analytes. The goal of this work is to study the retention and effective diffusion properties of porous graphitic carbon, to see if it is suitable for the wide injection bands and long run times associated with long, packed capillary-scale separations. The retention of a set of standard metabolites was studied for both stationary phases over a wide range of mobile phase conditions. This data showed that porous graphitic carbon benefits from significantly increased retention (often >100 fold) under initial gradient conditions for these metabolites, suggesting much improved ability to focus a wide injection band at the column inlet. The effective diffusion properties of these columns were studied using peak-parking experiments with the standard metabolites under a wide range of retention conditions. Under the high retention conditions, which can be associated with retention after injection loading for gradient separations, Deff/Dm∼0.1 for both the C18-bonded and porous graphitic carbon columns. As C18 bonded particles are widely, and successfully utilized for long gradient separations without issue of increasing peak width from longitudinal diffusion, this suggests that porous graphitic carbon should be amenable for long runtime gradient separations as well.

Combined solvent- and non-uniform temperature-programmed gradient liquid chromatography. I - A theoretical investigation

An new class of gradient liquid chromatography (GLC) is proposed and its performance is analyzed from a theoretical viewpoint. During the course of such gradients, both the solvent strength and the column temperature are simultaneously changed in time and space. The solvent and temperature gradients propagate along the chromatographic column at their own and independent linear velocity. This class of gradient is called combined solvent- and temperature-programmed gradient liquid chromatography (CST-GLC). The general expressions of the retention time, retention factor, and of the temporal peak width of the analytes at elution in CST-GLC are derived for linear solvent strength (LSS) retention models, modified van't Hoff retention behavior, linear and non-distorted solvent gradients, and for linear temperature gradients. In these conditions, the theory predicts that CST-GLC is equivalent to a unique and apparent dynamic solvent gradient. The apparent solvent gradient steepness is the sum of the solvent and temperature steepness. The apparent solvent linear velocity is the reciprocal of the steepness-averaged sum of the reciprocal of the actual solvent and temperature linear velocities. The advantage of CST-GLC over conventional GLC is demonstrated for the resolution of protein digests (peptide mapping) when applying smooth, retained, and linear acetonitrile gradients in combination with a linear temperature gradient (from 20°C to 90°C) using 300μm×150mm capillary columns packed with sub-2 μm particles. The benefit of CST-GLC is demonstrated when the temperature gradient propagates at the same velocity as the chromatographic speed. The experimental proof-of-concept for the realization of temperature ramps propagating at a finite and constant linear velocity is also briefly described.

General theory of peak compression in liquid chromatography

A new and general expression of the peak compression factor in liquid chromatography is derived. It applies to any type of gradients induced by non-uniform columns (stationary) or by temporal variations (dynamic) of the elution strength related to changes in solvent composition, temperature, or in any external field. The new equation is validated in two ideal cases for which the exact solutions are already known. From a practical viewpoint, it is used to predict the achievable degree of peak compression for curved retention models, retained solvent gradients, and for temperature-programmed liquid chromatography. The results reveal that: (1) curved retention models affect little the compression factor with respect to the best linear strength retention models, (2) gradient peaks can be indefinitely compressed with respect to isocratic peaks if the propagation speed of the gradient (solvent or temperature) becomes smaller than the chromatographic velocity, (3) limitations are inherent to the maximum intensity of the experimental intrinsic gradient steepness, and (4) dynamic temperature gradients can be advantageously combined to solvent gradients in order to improve peak capacities of microfluidic separation devices.

Wide injection zone compression in gradient reversed-phase liquid chromatography

Chromatographic zone broadening is a common issue in microfluidic chromatography, where the sample volume introduced on column often exceeds the column void volume. To better understand the propagation of wide chromatographic zones on a separation device, a series of MS Excel spreadsheets were developed to simulate the process. To computationally simplify these simulations, we investigated the effects of injection related zone broadening and its gradient related zone compression by tracking only the movements of zone boundaries on column. The effects of sample volume, sample solvent, gradient slope, and column length on zone broadening were evaluated and compared to experiments performed on 0.32mm I.D. microfluidic columns. The repetitive injection method (RIM) was implemented to generate experimental chromatograms where large sample volume scenarios can be emulated by injecting two discrete small injection plugs spaced in time. A good match between predicted and experimental RIM chromatograms was observed. We discuss the performance of selected retention models on the accuracy of predictions and use the developed spreadsheets for illustration of gradient zone focusing for both small molecules and peptides.

Repetitive injection method: a tool for investigation of injection zone formation and its compression in microfluidic liquid chromatography

Sample introduction in microfluidic liquid chromatography often generates wide zones rather than peaks, especially when a large sample volume (relative to column volume) is injected. Formation of wide injection zones can be further amplified when the sample is dissolved in a strong eluent. In some cases sample breakthrough may occur, especially when the injection is performed into short trapping columns. To investigate the band formation and subsequent zone focusing under gradient elution in situations such as these, we developed the Repetitive Injection Method (RIM), based on the temporally resolved introduction of two discrete peaks to a column, mimicking both the leading and trailing edges of a larger, singly injected sample zone. Using titanium microfluidic 0.32 mm I.D. columns, the results of RIM experiments were practically identical to injection of a correspondingly larger single zone volume. It was also experimentally shown that zone width (spacing between two injected peaks) decreases during gradient elution. We utilized RIM experiments to investigate wide sample zones created by strong sample solvent, and subsequent gradient zone focusing for a series of compounds. This experimental work was compared with computationally simulated chromatograms. The success of sample focusing during injection and gradient elution depends not only on an analyte's absolute retention, but also on how rapidly the analyte's retention changes during the mobile phase gradient.

Stationary phase characterization and method development

The properties of stationary phases and their characterization methods are reviewed. New and significant developments have occurred in the last few years, and new methods for stationary phase characterization have become available. The characterization methods are discussed, and the differences between the different methods are pointed out. In addition, method development approaches are reviewed, with special emphasis on recent developments that employ multiple parameters in parallel. Also, the renewed interest of temperature as a tool in method development is surveyed.

Selectivity in reversed-phase separations: General influence of solvent type and mobile phase pH

The influence of the mobile phase on retention is studied in this paper for a group of over 70 compounds with a broad range of multiple functional groups. We varied the pH of the mobile phase (pH 3, 7, and 10) and the organic modifier (methanol, acetonitrile (ACN), and tetrahydrofuran (THF)), using 15 different stationary phases. In this paper, we describe the overall retention and selectivity changes observed with these variables. We focus on the primary effects of solvent choice and pH. For example, transfer rules for solvent composition resulting in equivalent retention depend on the packing as well as on the type of analyte. Based on the retention patterns, one can calculate selectivity difference values for different variables. The selectivity difference is a measure of the importance of the different variables involved in method development. Selectivity changes specific to the type of analyte are described. The largest selectivity differences are obtained with pH changes.