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 paper introduces an efficient analytic time-domain model for chromatographic peak shapes, incorporating physical parameters and heterogeneity effects, with improved fitting accuracy and insights into tailing reduction.

Contribution

It provides a novel stochastic model with a rigorous theoretical foundation, enabling direct peak fitting and analysis of heterogeneity effects on peak tailing.

Findings

The model achieves lower residual errors than standard functions.

Heterogeneity can reduce peak tailing under certain conditions.

A decoupling approximation for slow kinetics is validated.

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.