Dependence on effective saturation of numbers of singlet peaks in one- and two-dimensional separations

The average numbers of singlet peaks in one-dimensional (1D) and two-dimensional (2D) separations of randomly distributed peaks are predicted by statistical-overlap theory and compared against the effective saturation. The effective saturation is a recently introduced metric of peak crowding that is more practitioner-friendly than the usual metric, the saturation. The effective saturation absorbs the average minimum resolution of statistical-overlap theory, facilitating the comparison of 1D and 2D separations by traditional metrics of resolution and peak capacity. In this paper, singlet peaks are identified with maxima produced by a single mixture constituent. Their effective saturations are calculated from published equations for the average minimum resolution of 1D singlet peaks, and from equations derived here for the average minimum resolution of 2D singlet peaks. The fractions of peaks that are singlets in 1D and 2D separations are predicted by statistical-overlap theory as functions of saturation but are compared as functions of effective saturation. The two fractions differ by no more than 0.033 at any effective saturation between 0 and 6, when the distribution of peak heights is exponential and the edge effect is neglected. This result shows that 1D and 2D separations of randomly distributed peaks are about the same in their ability to separate singlet peaks as maxima, when assessed relative to effective saturation. Empirical equations in effective saturation are reported for the fractions of peaks that are singlets. It is argued that the effective saturation is a good metric for comparing separations having different average minimum resolutions.     

Dependence on saturation of average minimum resolution in two-dimensional statistical-overlap theory: peak overlap in saturated two-dimensional separations

A theory is proposed for the dependence on saturation of the average minimum resolution R(*) in point-process statistical-overlap theory for two-dimensional separations. Peak maxima are modelled by clusters of overlapping circles in hexagonal arrangements similar to close-packed layers. Such clusters exist only for specific circle numbers, but equations are derived that facilitate prediction of equivalent cluster properties for any number of circles. A metric is proposed for the average minimum resolution that separates two such clusters into two maxima. From this metric, the average minimum resolution of the two nearest-neighbor single-component peaks (SCPs)--one in each cluster--is calculated. Its value varies with the number of SCPs in both clusters. These resolutions are weighted by the probability that the two clusters contain the postulated numbers of SCPs and summed to give R(*), which decreases with increasing saturation. The dependence of R(*) on saturation is combined with a theory correcting the probability of overlap in a reduced square for boundary effects. The numbers of maxima in simulations of 75, 150, and 300 randomly distributed bi-Gaussians having exponential heights and aspect ratios of 1, 30, and 60 are compared to predictions. Excellent agreement between maxima numbers and theory is found at low and high saturation. Good estimates of the numbers of bi-Gaussians in simulations are calculated by fitting theory to numbers of maxima using least-squares regression. The theory is applied to mimicked GC x GCs of 93 compounds having many correlated retention times, with predictions that agree fairly well with maxima numbers.     

Capillary electrophoretic chiral separations using cyclodextrin additives: III. Peak resolution surfaces for ibuprofen and homatropine as a function of the pH and the concentration of β-cyclodextrin

Our recent extended peak resolution equation of capillary electrophoresis has been combined with the multiple equilibria-based electrophoretic mobility model of chiral separations to describe peak resolution as a function of the composition of the background electrolyte (pH and the β-cyclodextrin concentration) and a function of the operating variables (effective portion of the applied potential, dimensionless electroosmotic flow coefficient). Using the previously determined model parameters, the resolution surfaces were calculated for a Type I chiral separation (ibuprofen), and a Type III chiral separation (homatropine). In Type I separations resolution can be obtained only over a narrow pH range in the vicinity of the pK_a value, and above a minimum value, the concentration of β-cyclodextrin plays a lesser role. In Type III separations, the pH- and β-cyclodextrin concentration-dependent resolution surface has two lobes, on which the migration order of the enantiomers is opposite. This can be an advantage in trace component analysis. In both Type I and Type III separations, peak resolution varies strongly with the dimensionless electroosmotic flow coefficient when its value is changed in the − 1 to 1 range. The loci of the pH-dependent and the β-cyclodextrin concentration-dependent resolution maxima do not shift significantly when the dimensionless electroosmotic flow coefficient is changed. This fact provides the analyst with an additional resolution enhancement tool that does not alter the selectivity of the separation. The utility of the model and its theoretical predictions has been demonstrated by comparing measured and calculated R_s values for ibuprofen and homatropine.     

Verification of statistical-overlap theory in micellar electrokinetic chromatography

The limited peak capacity of neutral compounds in micellar electrokinetic chromatography (MEKC) causes peak overlap in a simple 38-compound sample that is predicted by statistical-overlap theory (SOT). The low-concentration sample was prepared in-house from several compound classes to span the entire migration-time range and was resolved partially in a pH=7 phosphate buffer containing 50 mM sodium dodecyl sulfate. Peaks, singlets, doublets, and other multiplets were identified on the basis of known migration times and were counted at 13 voltages spanning 4 – 26 kV. These numbers agreed well with predictions of a simple SOT based on the assumption of an inhomogeneous Poisson distribution of migration times. Because the dispersion theory of MEKC is simple, the standard deviations of single-component peaks were modeled theoretically. As part of a new way to implement SOT, probability distributions of the numbers of peaks, singlets, and so on, were computed by Monte Carlo simulation. These distributions contain all theoretical information on peak multiplicity predictable by SOT and were used to evaluate the agreement between experiment and theory. The peak capacity of MEKC was calculated numerically and substituted into the simplest equations in SOT, affirming that peak overlap arises from limited peak capacity.     

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 β) is shown to generate more uniform chromatograms. Large scale range chromatograms (large β) 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.     

Comparison of mass resolution criteria in mass spectrometry

For a single peak, mass spectral resolution can be expressed in terms of peak width or ratio of peak position to peak width. Alternatively, for two equally intense peaks of equal width, resolution can be defined as the minimum peak separation such that the height of the valley between the combined peaks is less than a specified ratio (1%, 10%, 50%, 100%) of the individual (or combined) peak maximum. All these definitions depend on peak shape. Conversion formulae between various mass resolution criteria are presented for each of eight spectral peak shapes: Gaussian, triangular, trapezoidal, Lorentzian (absorption-mode, magnitude-mode, and sine-apodized magnitude-mode), and sinc (absorption-mode and magnitude-mode). From these formulae, mass resolutions based upon different criteria are readily compared for the same or different line shapes.     

Net analyte signal as a deconvolution-oriented resolution criterion in the optimisation of chromatographic techniques

The performance of two multivariate calibration measurements, multivariate selectivity (SEL(s)) and scalar net analyte signal (scalar NAS), as chromatographic objective functions (COFs), was investigated. Since both assessments are straightforwardly related to the quantification of analytes in the presence of interferents, they were expected to confer new features in the optimisation of compound resolution, not present in conventional assessments. These capabilities are especially interesting in situations of low resolution, where peak deconvolution becomes an attractive alternative. For comparison purposes, chromatographic resolution (R(s)) and peak purity (p(s)) were used as reference COFs. In order to correlate COFs with the probability of deconvolution error, an artificial peak crossing was used to generate 73 different peak arrangements, which were deconvolved using three different methods. SEL(s) exhibited the best correlation, which allowed predicting properly the risk of obtaining inaccurate deconvolutions. The optimisation of a poorly resolved mixture of 16 aromatic compounds by reversed-phase liquid chromatography with methanol-water and acetonitrile-water mobile phases was examined to investigate the differences in performance among the resolution criteria. In situations like these, SEL(s) tends to consider acceptable mobile phase compositions with partial coelution, which permits however the deconvolution with low errors. In contrast, p(s) selects compositions where the resolution of some compounds is sacrificed to enhance the separation of others. Scalar NAS was not so favourable as expected, since it depends on sampling frequency and peak widening. SEL(s) was not affected by these factors.     

Gas chromatography of 209 polychlorinated biphenyl congeners on an extremely efficient nonselective capillary column

The gas chromatographic–mass spectrometric (GC–MS) separation of all 209 polychlorinated biphenyl (PCB) congeners was studied on an extremely efficient 80 m × 0.1 mm i.d. capillary column coated with a 0.1 μm film of poly(5%-phenyl methyl)siloxane stationary phase. The quality of the separation and the number of resolved and coeluting peaks were compared to predictions according to the statistical overlap theory (SOT) and to literature data on PCB separations obtained by one-dimensional and comprehensive two-dimensional GC (GC × GC) and GC–MS. Mass spectral and chemometric deconvolution procedures were used to resolve overlapping peaks. On the highly efficient column, 195 PCB congeners were resolved in 96 min separation time using spectral and chemometric deconvolution. This number is comparable to the best separations described in GC × GC–MS mode. The novel method was developed for spectral deconvolution of overlapped PCB congeners which was verified determining the most toxic, dioxin-like PCBs both in the model mixture of 209 PCBs as well as in the Aroclor 1242 and Aroclor 1254 formulations.     

Comprehensive two-dimensional liquid chromatography — practical impacts of theoretical considerations. A review

A theory of comprehensive two-dimensional separations by liquid chromatographic techniques is overviewed. It includes heart-cutting and comprehensive two-dimensional separation modes, with attention to basic concepts of two-dimensional separations: resolution, peak capacity, efficiency, orthogonality and selectivity. Particular attention is paid to the effects of sample structure on the retention and advantages of a multi-dimensional HPLC for separation of complex samples according to structural correlations. Optimization of 2D separation systems, including correct selection of columns, flow-rate, fraction volumes and mobile phase, is discussed. Benefits of simultaneous programmed elution in both dimensions of LCxLC comprehensive separations are shown.     

Peak compression and resolution for electrophoretic separations in diverging microchannels

We report the results of experiments and simulations on electrokinetic flow in diverging microchannels (with cross-sectional area that increases with distance along the channel). Because of conservation of mass and charge, the velocity of an analyte in the channel decreases as the channel cross-section increases. Consequently, the leading edge of a band of sample moves more slowly than the trailing edge and the sample band is compressed. Sample peak widths, rather than increasing diffusively with time, can then be controlled by the geometry of the channel and can even be made to decrease with time. We consider the possibility of using this peak compression effect to improve the resolution of electrophoretic separations. Our results indicate that for typical separations that are dispersion limited, this peak compression effect is more than offset by the decreased distance between peaks, and the separation resolution in diverging channels is worse than that found for straight channels at the same applied voltage. For separations in very short channels or at very high field strengths, however, when the separation efficiency is injection limited, the peak compression effect is dominant and diverging channels can then be used to achieve improved separation resolution.     

Polymerisation and surface modification of methacrylate monoliths in polyimide channels and polyimide coated capillaries using 660 nm light emitting diodes

An investigation into the preparation of monolithic separation media utilising a cyanine dye sensitiser/triphenylbutylborate/N-methoxy-4-phenylpyridinium tetrafluoroborate initiating system activated by 660 nm light emitting diodes is reported. The work demonstrates multiple uses of red-light initiated polymerisation in the preparation of monolithic stationary phases within polyimide and polyimide coated channels and the modification of monolithic materials with molecules which absorb strongly in the UV region. This initiator complex was used to synthesise poly(butyl methacrylate-co-ethylene dimethacrylate) and poly(methyl methacrylate-co-ethylene dimethacrylate) monolithic stationary phases in polyimide coated fused silica capillaries of varying internal diameters, as well as within polyimide micro-fluidic chips. The repeatability of the preparation procedure and resultant monolithic structure was demonstrated with a batch of poly(butyl methacrylate-co-ethylene dimethacrylate) monoliths in 100 μm i.d. polyimide coated fused silica capillary, which were applied to the separation of a model protein mixture (ribonuclease A, cytochrome C, myoglobin and ovalbumin). Taking an average from 12 chromatograms originating from each batch, the maximum relative standard deviation of the retention factor (k) for the protein separations was recorded as 0.53%, the maximum variance for the selectivity factor (α) was 0.40% while the maximum relative standard deviation in peak resolution was 8.72%. All maxima were recorded for the Ribonuclease A/Cytochrome C peaks. Scanning electron microscopy confirmed the success of experiments in which poly(butyl methacrylate-co-ethylene dimethacrylate) monoliths were prepared using the same initiation approach in capillary and micro-fluidic chips, respectively. The initiating system was also applied to the photo-initiated grafting of a chromophoric monomer onto poly(butyl methacrylate-co-ethylene dimethacrylate) monoliths within poly(tetrafluoroethylene) coated fused silica capillaries.     

Fast hydrophilic interaction liquid chromatographic separations on bonded zwitterionic stationary phase

Separation science is an art of obtaining adequate resolution of the desired compounds in minimum time, and with minimum effort in terms of sample preparation and data evaluation. In LC, where selectivity is a main driving force for separation, the availability of different separation modes capable of operating at high flow rates is a way to make combined optimal use of selectivity, efficiency, and speed. The separation of polar and hydrophilic compounds is problematic in RP LC due to the poor retention. Hydrophilic interaction liquid chromatography (HILIC) is a more straightforward separation mode to address this problem. Herein, it is shown that separations in HILIC mode are equally efficient as for RP, providing a potential for very fast separations on short columns. This is not only facilitated by the low viscosity of the mobile phase compositions used, compared to typical RP eluents, but also due to higher column permeability. To exemplify this, baseline separations of uracil and cytosine are shown in less than 4 s and of Tamiflu and its main metabolite in less than 40 s, both under isocratic conditions. HILIC must therefore be considered having potential for high throughput purposes, and being an attractive candidate as the second separation dimension in 2-D HPLC.     

Development of a resolution metric for comprehensive two-dimensional chromatography

A new resolution metric for two-dimensional chromatography is proposed and tested. This resolution measurement is based on the concept of the (one-dimensional) valley-to-peak ratio, which has been adapted and modified for two-dimensional chromatography. Two questions are considered related to the computation of the resolution of a given (two-dimensional) peak. First, the concept of peak neighbourhood is revised, since it changes drastically from one- to two-dimensional chromatography. In a chromatogram resulting from a two-dimensional analysis, one peak may be surrounded by more than two neighbouring peaks. However, the neighbouring peaks can be remote from the peak or some interfering peaks may be in between. In these cases, it is not meaningful to compute the resolution between them. A method is proposed to determine whether a resolution measurement between two two-dimensional peaks is reasonable. Second, a measurement of the valley-to-peak ratio in two-dimensional chromatography is proposed. The measurement is based on the concept of the saddle point (which is defined for two-dimensional surface plots). A study of the correlation of the valley-to-peak ratio with the error obtained for quantification is presented. The new metric can be used as an estimator of the quantification errors. Also, valley-to-peak ratios can be calculated for one or more target peak(s) to estimate the separation quality of the entire chromatogram. This makes the proposed measurement suitable for optimisation purposes. Although the algorithm was developed for GC x GC, preliminary studies suggested that its application to other two-dimensional separation methods (e.g. LC x LC) should only require minor modification (if any).     

The impact of sampling time on peak capacity and analysis speed in on-line comprehensive two-dimensional liquid chromatography

Comprehensive two-dimensional liquid chromatography (2DLC) offers a number of practical advantages over optimized one-dimensional LC in peak capacity and thus in resolving power. The traditional “product rule” for overall peak capacity for a 2DLC system significantly overestimates peak capacity because it neglects under-sampling of the first dimension separation. Here we expand on previous work by more closely examining the effects of the first dimension peak capacity and gradient time, and the second dimension cycle times on the overall peak capacity of the 2DLC system. We also examine the effects of re-equilibration time on under-sampling as measured by the under-sampling factor and the influence of molecular type (peptide vs. small molecule) on peak capacity. We show that in fast 2D separations (less than 1 h), the second dimension is more important than the first dimension in determining overall peak capacity and conclude that extreme measures to enhance the first dimension peak capacity are usually unwarranted. We also examine the influence of sample types (small molecules vs. peptides) on second dimension peak capacity and peak capacity production rates, and how the sample type influences optimum second dimension gradient and re-equilibration times.     

The use of derivatives for establishing integration limits of chromatographic peaks

Summary The paper deals with composite peaks in which the resolution is not sufficient to allow simple area determinations with conventional integrator procedures. It is proposed to use the second derivatives of composite peaks, since the derivatives accentuate envelope perturbations due to overlapped peaks. In particular, when there are two solutes in the composite, and when the peak separation is between 2 and 4σ, the second derivative of the composite has two minima and three maxima. The second maximum is indicative of the cross point of the two solutes. This point can be used to initiate and/or terminate the integration of the components in the composite. Similarly, the second minimum occurs at a point close to the true maximum of the second peak in the composite. This point can also be used for the quantitative determination of the second component in the composite. The second derivative traces can also be integrated, but their utility in quantitative analysis of the peaks is questionable. An inversion procedure is given in which the second derivative trace is inverted to yield a trace similar to the conventional chromatograms but with better apparent resolution. In special circumstances, the inverted derivatives can be used for integration purposes.     

Statistical-overlap theory for elliptical zones of high aspect ratio in comprehensive two-dimensional separations

The low prediction by statistical-overlap theory of the numbers of singlets and peaks in two-dimensional separations containing zones represented by either circles of small number or eccentric ellipses of any number is shown to result from use of probability expressions for unbound spaces of infinite extent. An exact theory is derived for the probability of singlet formation in a reduced two-dimensional space of unit length, width, and area. The probability is a weighted sum of the probabilities of singlet formation in the interior, edge, and corner regions of the space, which depend only on saturation. The weighting factors are the fractions of area associated with each region and depend on the number of zones, the aspect ratio, the saturation, and the ellipse's spatial orientation. The average numbers of doublets, triplets, and peaks in the space are approximated by combining these results with Roach's equations describing the clustering of circles in an unbound two-dimensional space. Simulations show that theory predicts the number of singlets, doublets, triplets, and peaks, when the number of zones is 25 or more, the aspect ratio is 100 or less, and the saturation is 2 or less. The relationship is derived between the aspect ratios of ellipses in the reduced space and actual separation space. Calculations are presented for comprehensive two-dimensional gas chromatography.     

Oil-In-Water Microemulsion LC Determination of Pharmaceuticals Using Gradient Elution

Efficient and novel oil-in-water microemulsion HPLC (MELC) separations of a range of solutes have been performed on conventional reversed-phase HPLC columns using gradient elution. This work follows previous successful separations using isocratic oil-in-water MELC [1]. It was found that by changing certain variables, peak-peak resolution, separation selectivity, efficiency and solute retention could be manipulated. The method was compatible with very low UV detection wavelengths. A robust separation method was developed for the quantitative analysis of 2 steroids in a combination-inhaled product for asthma. The method offered similar chromatography and run time when compared with conventional HPLC modes, thus demonstrating its potential for routine use. Stability-indicating methods were developed to separate synthetic and degradative impurities from the main component peaks in 4 pharmaceutical products. The methods offered quicker analysis times and equivalent selectivity to conventional HPLC modes. In developing the separations the effect on the chromatography of varying the operating parameters was studied.     

Optimizing separations in reversed-phase liquid chromatography by varying pH and solvent composition

Summary An interpretive optimization procedure in which pH can be one of the variables is presented with the emphasis on optimizing separations. When varying the pH in reversed-phase liquid chromatography the retention of ionogenic solutes will change. Thus, the selectivity between ionogenic and neutral solutes or between ionogenic solutes mutually can be optimized. However, pH also greatly affects the efficiency (plate count) and peak shape (asymmetry). Optimum selectivity (i.e. large differences in retention times) may be observed under conditions where peaks are broad and asymmetrical. Thus, it is essential to simultaneously consider retention, peak width and peak shape and their effects on separation (effective resolution) in pH-optimization studies. A procedure in which this is done is presented and applied to optimizing the separation of a synthetic mixture of selected pharmaceuticals. After initial experiments to establish the parameter space (boundaries for pH and binary methanol — water composition), twelve experiments are performed according to a 3×4 experimental design. At each loaction the retention, peak height, peak area and peak symmetry are recorded for each solute. These data are then used to build models for each of the four characteristics and for each solute. From this set of models the response surface, describing the quality of separation as a function of pH and composition, can be calculated. A variety of optimization criteria (quantifying quality of separation) can be used. The optimum corresponds to the highest point on the response surface.     

The Need for Two‐Dimensional Gas Chromatography: Extent of Overlap in One‐Dimensional Gas Chromatograms

The need for two‐dimensional gas chromatography is justified by the extent of peak overlap in one‐dimensional gas chromatograms (GCs) of complex mixtures. Such overlap was predicted long ago by statistical‐overlap theory (SOT). In this paper, SOT is conceptually reviewed and its predictions are shown to be quantitatively accurate. GCs of complex mixtures of polychlorinated biphenyls, pyridine‐ and nitrogen‐containing polynuclear aromatic hydrocarbons, tetrachlorodibenzo‐p‐dioxins and dibenzofurans, fatty acid methyl esters, flavors and fragrances, and naphtha were simulated by commercial GC software on DB‐1, DB‐5, and Stabilwax stationary phases. The numbers of peak maxima in the GCs agreed with predictions of SOT, when the interval of time between successive peaks of pure compounds was described by Poisson statistics. This agreement was realized even though the time intervals actually are deterministic, not statistical. In addition, the numbers of mixture components were predicted with accuracy by regression of peak numbers against SOT. Similar regressions have been reported before, but the theory used here is more sophisticated and its predictions consequently are more accurate. Future directions for finalizing SOT are suggested.     

Detector resolution required for accurate identification in common gamma-ray masking situations

Accurate nuclide identification depends on the ability to determine if specific peaks are present in the spectrum. Several current handheld nuclide identifiers and portal monitors use a variant of a peak quality value for this. The peak quality is usually calculated as the peak area divided by the uncertainty of the peak area and when this quotient is above a threshold value, the peak is said to be present. Other works [Terracol et al. In: 2004 IEEE Nuclear Science Symposium Conference Record, Rome, Italy, 2004, Ryder In: Scanning Electron Microscopy/1977 V. 1, Proceedings of the Workshop on Analytical Electron Microscopy, Chicago, 1977] have developed a formalism to calculate the peak uncertainty for interfering peaks based on the detector resolution, background, individual peak areas, and peak separation. The threshold on peak uncertainty determines the minimum activity that will be identified or detected. Care must be used in the selection of the threshold in order to comply with the false positive and false negative requirements of the detection system regime, or “concept of operations”. The performance standards for the handheld identifiers and portal monitors specify the nuclides required to be identified. From this list and other commonly expected nuclides, the energies of the expected gamma rays can be tallied, yielding a table of the separations of adjacent peaks possible in the collected spectrum. Using the formalism, the peak quality value can be determined as a function of the detector resolution, peak area and background for the energy separations in the table determined above. Results are shown for the cases of HEU and plutonium with the masking nuclides of NORM, ¹³³Ba, or ⁵⁷Co for both germanium and sodium iodide detectors. Typical resolutions, efficiencies and counting times were used.