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Chromatographic Peak Shape from Stochastic Model: Analytic Time-Domain Expression in Terms of Physical Parameters and Conditions under which Heterogeneity Reduces Tailing

Hernán R. Sánchez

TL;DR

This work develops a stochastic–diffusive framework to predict chromatographic peak shapes in isocratic, isothermal conditions by linking the single-particle residence-time PDF to a time-domain peak expression. It couples advective–diffusive transport (with axial dispersion and finite injection variance) to a two-mechanism retention process (fast and slow), yielding a closed-form peak description that combines a Gaussian mobile-plus-fast component with a Poisson–Gamma slow component; under high-Péclet conditions, the fast part converges to a Gaussian, enabling efficient peak fitting via moment-matched surrogates. A rigorous foundation is provided for the Poisson assumption and its excess-variance due to rate fluctuations, and a statistical interpretation of HETP relates macroscopic plate counts to microscopic event counts, yielding a conservative lower bound on retention events. The model’s convolution-based final density is expressed in terms of confluent hypergeometric functions and is validated against experimental peaks, showing superior residual errors compared with standard EMG-based models. Importantly, mechanistic heterogeneity need not worsen tailing; under certain parameter regimes, the presence of two retention mechanisms can reduce peak asymmetry, with transport and fast-kinetic effects further expanding this favorable region.

Abstract

A time-domain representation of chromatographic peak shapes is presented as an analytic expression designed for high computational efficiency, which can be used for direct time-domain peak fitting with parameters that represent physical quantities. The underlying model integrates the effects of axial diffusion (molecular and multipath/eddy), finite initial spatial variance, and two distinct retention mechanisms: one characterized by a high rate of short-duration events (fast kinetics), and another by a low rate of long-duration events (slow kinetics). Fits to experimental chromatograms yield substantially smaller residual standard error (RSE) than the standard EMG and the lowest average normalised RSE among 12 established peak-shape functions in the examined cases. The stochastic approach is reformulated using single-particle probability laws, providing a rigorous basis for future theoretical extensions. The validity of the foundational Poisson assumption is critically examined by deriving expressions for the excess variance caused by correlated microscopic retention rate fluctuations. A statistical interpretation of the HETP is presented and used to determine a lower bound on the number of microscopic retention events from chromatogram-derived macroscopic observables. This, in turn, justifies the applicability of the Gaussian limit for the mobile-plus-fast component, as established by analysis of the cumulant generating function of the closed-form benchmark derived herein. The contribution from the slow-kinetic mechanism is incorporated via a decoupling approximation, whose validity is established through a cumulant-based analysis that explicitly bounds the decoupling-induced error. Finally, the notion that mechanistic heterogeneity necessarily exacerbates peak tailing is qualified, analytically delineating parameter regions in which it leads to a reduction in peak asymmetry.

Chromatographic Peak Shape from Stochastic Model: Analytic Time-Domain Expression in Terms of Physical Parameters and Conditions under which Heterogeneity Reduces Tailing

TL;DR

This work develops a stochastic–diffusive framework to predict chromatographic peak shapes in isocratic, isothermal conditions by linking the single-particle residence-time PDF to a time-domain peak expression. It couples advective–diffusive transport (with axial dispersion and finite injection variance) to a two-mechanism retention process (fast and slow), yielding a closed-form peak description that combines a Gaussian mobile-plus-fast component with a Poisson–Gamma slow component; under high-Péclet conditions, the fast part converges to a Gaussian, enabling efficient peak fitting via moment-matched surrogates. A rigorous foundation is provided for the Poisson assumption and its excess-variance due to rate fluctuations, and a statistical interpretation of HETP relates macroscopic plate counts to microscopic event counts, yielding a conservative lower bound on retention events. The model’s convolution-based final density is expressed in terms of confluent hypergeometric functions and is validated against experimental peaks, showing superior residual errors compared with standard EMG-based models. Importantly, mechanistic heterogeneity need not worsen tailing; under certain parameter regimes, the presence of two retention mechanisms can reduce peak asymmetry, with transport and fast-kinetic effects further expanding this favorable region.

Abstract

A time-domain representation of chromatographic peak shapes is presented as an analytic expression designed for high computational efficiency, which can be used for direct time-domain peak fitting with parameters that represent physical quantities. The underlying model integrates the effects of axial diffusion (molecular and multipath/eddy), finite initial spatial variance, and two distinct retention mechanisms: one characterized by a high rate of short-duration events (fast kinetics), and another by a low rate of long-duration events (slow kinetics). Fits to experimental chromatograms yield substantially smaller residual standard error (RSE) than the standard EMG and the lowest average normalised RSE among 12 established peak-shape functions in the examined cases. The stochastic approach is reformulated using single-particle probability laws, providing a rigorous basis for future theoretical extensions. The validity of the foundational Poisson assumption is critically examined by deriving expressions for the excess variance caused by correlated microscopic retention rate fluctuations. A statistical interpretation of the HETP is presented and used to determine a lower bound on the number of microscopic retention events from chromatogram-derived macroscopic observables. This, in turn, justifies the applicability of the Gaussian limit for the mobile-plus-fast component, as established by analysis of the cumulant generating function of the closed-form benchmark derived herein. The contribution from the slow-kinetic mechanism is incorporated via a decoupling approximation, whose validity is established through a cumulant-based analysis that explicitly bounds the decoupling-induced error. Finally, the notion that mechanistic heterogeneity necessarily exacerbates peak tailing is qualified, analytically delineating parameter regions in which it leads to a reduction in peak asymmetry.

Paper Structure

This paper contains 16 sections, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Homogeneous model with deterministic mobile-phase residence time
  4. Distribution of the Number of Retention Events
  5. Residence–Time Distribution within the Column and Its Statistical Moments
  6. Tailing and mechanistic heterogeneity
  7. Time–domain peak expression obtained by coupling axial diffusion with stochastic retention
  8. A Statistical Foundation for the Height Equivalent to one Theoretical Plate
  9. Lower Bound for the Number of Retention Events from the Chromatogram
  10. Composite Distribution for Mobile Phase and Fast Retention
  11. Normal Approximation and non-spike injection
  12. Treatment of Infrequent and Slow-Release Events
  13. Final Composite Density
  14. Validation of the Functional Form against Experimental Data
  15. Tailing from Heterogeneous Slow Kinetics in a Comprehensive Transport Model
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Skewness ratio $\gamma_{AB}/\gamma$ versus release-rate ratio $r=\beta_B/\beta_A$ for selected fractions $q_A$. Values below unity indicate heterogeneity-induced reduction of tailing.
  • Figure 2: Graphical representation of least-squares fits obtained using the proposed function (blue line) and the EMG (red line) applied to experimental data (gray crosses). Panels A and C correspond to a cyclohexane peak from gas chromatography (see Ref. pap2001); Panels B and D correspond to a fluoride peak from ion chromatography (see Ref. thermoAN003548). Top panels (A–B): fitted curves. Bottom panels (C–D): residuals.