Capturing Curved Lag Plots in Dye Adsorption Kinetics Using a Generalized Three-State Discrete-Time Markov Chain Model 


Vol. 27,  No. 3, pp. 1381-1400, Mar.  2026
10.1007/s12221-025-01302-7


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  Abstract

Applying discrete-time Markov chain (DTMC) theory to model adsorption kinetics in textile exhaust dyeing introduces a novel methodological framework. However, the dual-state DTMC (DMC) model fails to capture the curvature of lag plots under large time steps. To address this limitation, we propose a generalized modeling approach—the three-state DTMC (TMC) model. The TMC model incorporates a three-state mechanism (free, transitional, and fully adsorbed) and employs a signed transition probability matrix to represent directional dynamics, thereby enabling more accurate modeling at coarser temporal resolutions. We validated the TMC model using five representative dye/fiber systems—reactive/cotton, direct/viscose, acid/wool, disperse/polyester, and basic/acrylic—covering a wide range of adsorption mechanisms and reversibility characteristics. Benchmarking against five classical kinetic models demonstrated that the TMC model outperforms existing approaches across multiple evaluation metrics, including adjusted R-squared, root-mean-square error, Bayesian information criterion, residual violin plots, and residual time-series analysis. By conceptualizing transfer regions as symmetric spaces, the TMC model avoids reliance on assumptions regarding rate-limiting steps, achieving universal applicability across diverse dyeing systems. Further analysis reveals that system equilibrium is determined exclusively by adsorption and desorption transition probabilities, which experimental conditions modulate to shape both dynamic evolution and steady-state distribution. By extending transition probabilities to signed values with directional meaning—where negatives indicate deviations from assumed transfer pathways—the TMC model retains the core structure of DMC while capturing complex adsorption dynamics. This work provides a unified framework for modeling textile dyeing kinetics and offers a practical extension of DTMC theory to dyeing processes.

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  Cite this article

[IEEE Style]

D. Lei and J. Huang, "Capturing Curved Lag Plots in Dye Adsorption Kinetics Using a Generalized Three-State Discrete-Time Markov Chain Model," Fibers and Polymers, vol. 27, no. 3, pp. 1381-1400, 2026. DOI: 10.1007/s12221-025-01302-7.

[ACM Style]

Dapeng Lei and Jianhua Huang. 2026. Capturing Curved Lag Plots in Dye Adsorption Kinetics Using a Generalized Three-State Discrete-Time Markov Chain Model. Fibers and Polymers, 27, 3, (2026), 1381-1400. DOI: 10.1007/s12221-025-01302-7.

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